Properties

Label 40-1169e20-1.1-c0e20-0-0
Degree $40$
Conductor $2.271\times 10^{61}$
Sign $1$
Analytic cond. $2.08656\times 10^{-5}$
Root an. cond. $0.763810$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 3·4-s − 2·6-s + 7-s + 2·8-s − 11-s − 3·12-s + 2·14-s + 16-s + 2·19-s − 21-s − 2·22-s − 2·24-s − 10·25-s − 27-s + 3·28-s + 2·29-s − 31-s + 33-s + 4·38-s − 2·42-s − 3·44-s − 47-s − 48-s + 49-s − 20·50-s + ⋯
L(s)  = 1  + 2·2-s − 3-s + 3·4-s − 2·6-s + 7-s + 2·8-s − 11-s − 3·12-s + 2·14-s + 16-s + 2·19-s − 21-s − 2·22-s − 2·24-s − 10·25-s − 27-s + 3·28-s + 2·29-s − 31-s + 33-s + 4·38-s − 2·42-s − 3·44-s − 47-s − 48-s + 49-s − 20·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 167^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 167^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(7^{20} \cdot 167^{20}\)
Sign: $1$
Analytic conductor: \(2.08656\times 10^{-5}\)
Root analytic conductor: \(0.763810\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1169} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 7^{20} \cdot 167^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6892798325\)
\(L(\frac12)\) \(\approx\) \(0.6892798325\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
167 \( ( 1 - T )^{20} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
3 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
5 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
11 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
13 \( ( 1 - T )^{20}( 1 + T )^{20} \)
17 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
23 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
29 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
37 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
41 \( ( 1 - T )^{20}( 1 + T )^{20} \)
43 \( ( 1 - T )^{20}( 1 + T )^{20} \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
53 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
59 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
67 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
71 \( ( 1 - T )^{20}( 1 + T )^{20} \)
73 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
79 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
83 \( ( 1 - T )^{20}( 1 + T )^{20} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.33958912540938755610289945719, −2.33732670246099255852485687764, −2.25861590778850089877773312935, −2.25601632833754889156856513200, −2.09929615802560002363857162863, −2.09416846308443063378137441369, −2.04944508678169887191389711688, −1.98634767357860390441959270933, −1.94364842905917029471732187292, −1.89971953395639871999706834811, −1.75056930022064598301881325937, −1.74413061951590468253205116935, −1.74164250543072201830035004843, −1.67193666376029192835402814649, −1.62141379930701244510171650078, −1.58083986465468025211141020362, −1.46577241375150951201930867310, −1.39433189456944800881750172532, −0.993782236146804785311703723286, −0.992770933656662698523989163583, −0.957728265793958044145439233526, −0.868664887390317409391949395980, −0.829853154545735253900893259609, −0.69008797659718670473038976226, −0.35010044936447848775972091567, 0.35010044936447848775972091567, 0.69008797659718670473038976226, 0.829853154545735253900893259609, 0.868664887390317409391949395980, 0.957728265793958044145439233526, 0.992770933656662698523989163583, 0.993782236146804785311703723286, 1.39433189456944800881750172532, 1.46577241375150951201930867310, 1.58083986465468025211141020362, 1.62141379930701244510171650078, 1.67193666376029192835402814649, 1.74164250543072201830035004843, 1.74413061951590468253205116935, 1.75056930022064598301881325937, 1.89971953395639871999706834811, 1.94364842905917029471732187292, 1.98634767357860390441959270933, 2.04944508678169887191389711688, 2.09416846308443063378137441369, 2.09929615802560002363857162863, 2.25601632833754889156856513200, 2.25861590778850089877773312935, 2.33732670246099255852485687764, 2.33958912540938755610289945719

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

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