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Properties

Label 88-1939e44-1.1-c0e44-0-1
Degree $88$
Conductor $4.502\times 10^{144}$
Sign $1$
Analytic cond. $0.235687$
Root an. cond. $0.983710$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 7-s + 9-s + 16-s + 23-s − 25-s + 28-s + 21·29-s + 36-s − 3·43-s + 49-s + 63-s + 67-s + 2·71-s + 79-s + 81-s + 92-s − 100-s + 3·107-s + 112-s + 2·113-s + 21·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 4-s + 7-s + 9-s + 16-s + 23-s − 25-s + 28-s + 21·29-s + 36-s − 3·43-s + 49-s + 63-s + 67-s + 2·71-s + 79-s + 81-s + 92-s − 100-s + 3·107-s + 112-s + 2·113-s + 21·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{44} \cdot 277^{44}\right)^{s/2} \, \Gamma_{\C}(s)^{44} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{44} \cdot 277^{44}\right)^{s/2} \, \Gamma_{\C}(s)^{44} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(88\)
Conductor: \(7^{44} \cdot 277^{44}\)
Sign: $1$
Analytic conductor: \(0.235687\)
Root analytic conductor: \(0.983710\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((88,\ 7^{44} \cdot 277^{44} ,\ ( \ : [0]^{44} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(22.60164940\)
\(L(\frac12)\) \(\approx\) \(22.60164940\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} \)
277 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
3 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
5 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} + T^{24} + T^{26} - T^{30} - T^{32} + T^{36} + T^{38} - T^{42} - T^{44} - T^{46} + T^{50} + T^{52} - T^{56} - T^{58} + T^{62} + T^{64} - T^{68} - T^{70} + T^{74} + T^{76} - T^{80} - T^{82} + T^{86} + T^{88} \)
11 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} )^{2} \)
17 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} + T^{24} + T^{26} - T^{30} - T^{32} + T^{36} + T^{38} - T^{42} - T^{44} - T^{46} + T^{50} + T^{52} - T^{56} - T^{58} + T^{62} + T^{64} - T^{68} - T^{70} + T^{74} + T^{76} - T^{80} - T^{82} + T^{86} + T^{88} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
29 \( ( 1 - T + T^{2} )^{22}( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
31 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} + T^{24} + T^{26} - T^{30} - T^{32} + T^{36} + T^{38} - T^{42} - T^{44} - T^{46} + T^{50} + T^{52} - T^{56} - T^{58} + T^{62} + T^{64} - T^{68} - T^{70} + T^{74} + T^{76} - T^{80} - T^{82} + T^{86} + T^{88} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} )^{2} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} + T^{19} + T^{20} + T^{21} + T^{22} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} )^{2} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} - T^{19} + T^{20} - T^{21} + T^{22} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{12} - T^{13} + T^{15} - T^{16} + T^{18} - T^{19} + T^{21} - T^{22} + T^{23} - T^{25} + T^{26} - T^{28} + T^{29} - T^{31} + T^{32} - T^{34} + T^{35} - T^{37} + T^{38} - T^{40} + T^{41} - T^{43} + T^{44} )( 1 + T - T^{3} - T^{4} + T^{6} + T^{7} - T^{9} - T^{10} + T^{12} + T^{13} - T^{15} - T^{16} + T^{18} + T^{19} - T^{21} - T^{22} - T^{23} + T^{25} + T^{26} - T^{28} - T^{29} + T^{31} + T^{32} - T^{34} - T^{35} + T^{37} + T^{38} - T^{40} - T^{41} + T^{43} + T^{44} ) \)
97 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} + T^{14} - T^{18} - T^{20} + T^{24} + T^{26} - T^{30} - T^{32} + T^{36} + T^{38} - T^{42} - T^{44} - T^{46} + T^{50} + T^{52} - T^{56} - T^{58} + T^{62} + T^{64} - T^{68} - T^{70} + T^{74} + T^{76} - T^{80} - T^{82} + T^{86} + T^{88} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{88} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.28205654826943338345622548425, −1.26571319807726438936356965737, −1.24861246426519845139624303320, −1.19815234605130504794971729720, −1.19652527179737471832733168344, −1.14201213898561561743752183569, −1.07983907886845868918752612739, −1.04318577182004639109203584119, −1.03993977031315537502071651773, −1.03754613807700890802363048419, −1.03012514805910334820959398121, −1.01692482813237454740534814658, −0.955770840677941600870898890498, −0.943978810654608095799014813966, −0.908198482560692869214063708316, −0.904598700444866168441821087845, −0.863836674481165987503557011613, −0.845130723419594480708623639305, −0.801029017853834154677458257605, −0.72285614926090624812031540122, −0.63086471063977031856843932013, −0.62609710414951423249881520008, −0.58836215154244432563753190519, −0.49995838411459922617032022188, −0.42102016831030734463321939896, 0.42102016831030734463321939896, 0.49995838411459922617032022188, 0.58836215154244432563753190519, 0.62609710414951423249881520008, 0.63086471063977031856843932013, 0.72285614926090624812031540122, 0.801029017853834154677458257605, 0.845130723419594480708623639305, 0.863836674481165987503557011613, 0.904598700444866168441821087845, 0.908198482560692869214063708316, 0.943978810654608095799014813966, 0.955770840677941600870898890498, 1.01692482813237454740534814658, 1.03012514805910334820959398121, 1.03754613807700890802363048419, 1.03993977031315537502071651773, 1.04318577182004639109203584119, 1.07983907886845868918752612739, 1.14201213898561561743752183569, 1.19652527179737471832733168344, 1.19815234605130504794971729720, 1.24861246426519845139624303320, 1.26571319807726438936356965737, 1.28205654826943338345622548425

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.