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Properties

Label 1-643-643.13-r1-0-0
Degree $1$
Conductor $643$
Sign $-0.201 + 0.979i$
Analytic cond. $69.0999$
Root an. cond. $69.0999$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.131 + 0.991i)2-s + (0.904 + 0.426i)3-s + (−0.965 + 0.261i)4-s + (0.978 − 0.204i)5-s + (−0.303 + 0.952i)6-s + (0.996 − 0.0782i)7-s + (−0.386 − 0.922i)8-s + (0.636 + 0.771i)9-s + (0.331 + 0.943i)10-s + (0.974 + 0.223i)11-s + (−0.984 − 0.175i)12-s + (0.448 − 0.893i)13-s + (0.208 + 0.977i)14-s + (0.972 + 0.232i)15-s + (0.863 − 0.504i)16-s + (−0.980 − 0.194i)17-s + ⋯
L(s)  = 1  + (0.131 + 0.991i)2-s + (0.904 + 0.426i)3-s + (−0.965 + 0.261i)4-s + (0.978 − 0.204i)5-s + (−0.303 + 0.952i)6-s + (0.996 − 0.0782i)7-s + (−0.386 − 0.922i)8-s + (0.636 + 0.771i)9-s + (0.331 + 0.943i)10-s + (0.974 + 0.223i)11-s + (−0.984 − 0.175i)12-s + (0.448 − 0.893i)13-s + (0.208 + 0.977i)14-s + (0.972 + 0.232i)15-s + (0.863 − 0.504i)16-s + (−0.980 − 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ -0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.744929637 + 3.366044970i\)
\(L(\frac12)\) \(\approx\) \(2.744929637 + 3.366044970i\)
\(L(1)\) \(\approx\) \(1.627002550 + 1.160208438i\)
\(L(1)\) \(\approx\) \(1.627002550 + 1.160208438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (0.131 + 0.991i)T \)
3 \( 1 + (0.904 + 0.426i)T \)
5 \( 1 + (0.978 - 0.204i)T \)
7 \( 1 + (0.996 - 0.0782i)T \)
11 \( 1 + (0.974 + 0.223i)T \)
13 \( 1 + (0.448 - 0.893i)T \)
17 \( 1 + (-0.980 - 0.194i)T \)
19 \( 1 + (-0.367 + 0.929i)T \)
23 \( 1 + (-0.941 - 0.335i)T \)
29 \( 1 + (0.294 + 0.955i)T \)
31 \( 1 + (-0.687 - 0.725i)T \)
37 \( 1 + (0.395 + 0.918i)T \)
41 \( 1 + (0.837 - 0.545i)T \)
43 \( 1 + (0.358 + 0.933i)T \)
47 \( 1 + (0.430 - 0.902i)T \)
53 \( 1 + (0.967 - 0.251i)T \)
59 \( 1 + (-0.0830 + 0.996i)T \)
61 \( 1 + (-0.843 + 0.537i)T \)
67 \( 1 + (0.996 + 0.0879i)T \)
71 \( 1 + (0.412 - 0.910i)T \)
73 \( 1 + (0.887 + 0.461i)T \)
79 \( 1 + (0.582 - 0.813i)T \)
83 \( 1 + (-0.687 + 0.725i)T \)
89 \( 1 + (0.924 - 0.381i)T \)
97 \( 1 + (-0.768 - 0.640i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.86442348717892898387876769155, −21.649688484712025545473096824580, −20.84455230132187327062566871674, −19.99303731618337537127810221870, −19.35936391564159992248238663885, −18.39638961008266497909343430048, −17.80408718884241749209910923186, −17.15235994684856568194596139567, −15.4720038584220973449322254750, −14.3616827437091691016920290420, −14.08544425668088072440842455786, −13.385532694031462682602149915233, −12.42431498138852600891650253193, −11.41803633395791610208530309851, −10.74326169432789515698036341290, −9.39227254909364961796310593082, −9.08728502152570993284939688531, −8.19939068075763377834275056253, −6.828780600955115162587723630048, −5.89927122488637153534994717908, −4.51013094334578556668263411784, −3.81418613925755777370845700338, −2.3788074385826665941565406928, −1.95021115170858071660829107508, −1.006388197674036293516924942270, 1.205526359690438021088444625322, 2.31200459028436480376280668383, 3.78863532243351815796448681830, 4.5220117056768993264605582333, 5.48408586493784952023171396886, 6.41969091721208152274016532557, 7.55992565136458545833903373587, 8.436668823575084440176462204243, 8.98560918539944703729338265468, 9.90561322319049324160847862644, 10.76567448626598223168290284023, 12.33753047327606762703778558142, 13.29260961730479160990362955129, 13.97327893862649894174081140701, 14.62005352629087574589469352198, 15.19212346114742107086057800489, 16.31295562895875754242711488750, 17.000197199594365636304782549611, 17.97200454162800875077209874595, 18.38515335270733270672470784327, 19.86626299615908700063167443568, 20.53305851625986869077098384351, 21.40282052410590534166604298558, 22.07011157180246545739540618566, 22.81954870879231106637562915176

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