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Properties

Label 18-3463e9-3463.3462-c0e9-0-0
Degree $18$
Conductor $7.163\times 10^{31}$
Sign $1$
Analytic cond. $137.558$
Root an. cond. $1.31463$
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  − 2-s − 7-s + 9·9-s + 14-s − 9·18-s + 9·25-s − 31-s − 9·50-s − 61-s + 62-s − 9·63-s − 67-s − 71-s − 73-s − 79-s + 45·81-s − 83-s − 89-s − 109-s + 9·121-s + 122-s + 9·126-s + 127-s + 131-s + 134-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2-s − 7-s + 9·9-s + 14-s − 9·18-s + 9·25-s − 31-s − 9·50-s − 61-s + 62-s − 9·63-s − 67-s − 71-s − 73-s − 79-s + 45·81-s − 83-s − 89-s − 109-s + 9·121-s + 122-s + 9·126-s + 127-s + 131-s + 134-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3463^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3463^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(3463^{9}\)
Sign: $1$
Analytic conductor: \(137.558\)
Root analytic conductor: \(1.31463\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3463} (3462, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 3463^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.556138257\)
\(L(\frac12)\) \(\approx\) \(6.556138257\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3463 \( 1+O(T) \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
3 \( ( 1 - T )^{9}( 1 + T )^{9} \)
5 \( ( 1 - T )^{9}( 1 + T )^{9} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
11 \( ( 1 - T )^{9}( 1 + T )^{9} \)
13 \( ( 1 - T )^{9}( 1 + T )^{9} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( ( 1 - T )^{9}( 1 + T )^{9} \)
23 \( ( 1 - T )^{9}( 1 + T )^{9} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
37 \( ( 1 - T )^{9}( 1 + T )^{9} \)
41 \( ( 1 - T )^{9}( 1 + T )^{9} \)
43 \( ( 1 - T )^{9}( 1 + T )^{9} \)
47 \( ( 1 - T )^{9}( 1 + T )^{9} \)
53 \( ( 1 - T )^{9}( 1 + T )^{9} \)
59 \( ( 1 - T )^{9}( 1 + T )^{9} \)
61 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
67 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
71 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
73 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
79 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
83 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
89 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
97 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.44549646192911275585946772883, −3.39270623965460111597278534165, −3.34410603163154223314176028463, −3.21641672255685690904525702137, −3.10671358907958503356796142769, −2.86214065203579897208310258632, −2.73147087354202640082867422159, −2.72400777294142610406254114229, −2.68282154583767999372604435424, −2.62876939608473007376433782566, −2.35154631391521023062478218473, −2.00517632622879473526237045753, −1.97758467354071707744966441615, −1.90286456366073024356308082879, −1.88636200164215538576309938133, −1.70233209911403223105037985360, −1.60974567610024042755905783411, −1.45780866639496573855938398120, −1.36809670798334478821458590639, −1.17753220348956358631624719648, −1.06899684975917550550953723983, −1.00292032347397514256434972495, −0.840089581728883988942120886055, −0.829246120126735184682107095608, −0.74518395901482590680272071845, 0.74518395901482590680272071845, 0.829246120126735184682107095608, 0.840089581728883988942120886055, 1.00292032347397514256434972495, 1.06899684975917550550953723983, 1.17753220348956358631624719648, 1.36809670798334478821458590639, 1.45780866639496573855938398120, 1.60974567610024042755905783411, 1.70233209911403223105037985360, 1.88636200164215538576309938133, 1.90286456366073024356308082879, 1.97758467354071707744966441615, 2.00517632622879473526237045753, 2.35154631391521023062478218473, 2.62876939608473007376433782566, 2.68282154583767999372604435424, 2.72400777294142610406254114229, 2.73147087354202640082867422159, 2.86214065203579897208310258632, 3.10671358907958503356796142769, 3.21641672255685690904525702137, 3.34410603163154223314176028463, 3.39270623965460111597278534165, 3.44549646192911275585946772883

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

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