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Properties

Label 1-80e2-6400.477-r1-0-0
Degree $1$
Conductor $6400$
Sign $0.998 - 0.0559i$
Analytic cond. $687.775$
Root an. cond. $687.775$
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  + (−0.664 + 0.747i)3-s + (0.980 − 0.195i)7-s + (−0.117 − 0.993i)9-s + (−0.842 + 0.539i)11-s + (0.984 + 0.175i)13-s + (−0.972 + 0.233i)17-s + (0.603 − 0.797i)19-s + (−0.505 + 0.862i)21-s + (0.962 − 0.271i)23-s + (0.820 + 0.571i)27-s + (0.0588 + 0.998i)29-s + (0.156 − 0.987i)31-s + (0.156 − 0.987i)33-s + (−0.0196 − 0.999i)37-s + (−0.785 + 0.619i)39-s + ⋯
L(s)  = 1  + (−0.664 + 0.747i)3-s + (0.980 − 0.195i)7-s + (−0.117 − 0.993i)9-s + (−0.842 + 0.539i)11-s + (0.984 + 0.175i)13-s + (−0.972 + 0.233i)17-s + (0.603 − 0.797i)19-s + (−0.505 + 0.862i)21-s + (0.962 − 0.271i)23-s + (0.820 + 0.571i)27-s + (0.0588 + 0.998i)29-s + (0.156 − 0.987i)31-s + (0.156 − 0.987i)33-s + (−0.0196 − 0.999i)37-s + (−0.785 + 0.619i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $0.998 - 0.0559i$
Analytic conductor: \(687.775\)
Root analytic conductor: \(687.775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6400} (477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6400,\ (1:\ ),\ 0.998 - 0.0559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.064159682 - 0.05776987105i\)
\(L(\frac12)\) \(\approx\) \(2.064159682 - 0.05776987105i\)
\(L(1)\) \(\approx\) \(0.9809709062 + 0.1599581392i\)
\(L(1)\) \(\approx\) \(0.9809709062 + 0.1599581392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.664 + 0.747i)T \)
7 \( 1 + (0.980 - 0.195i)T \)
11 \( 1 + (-0.842 + 0.539i)T \)
13 \( 1 + (0.984 + 0.175i)T \)
17 \( 1 + (-0.972 + 0.233i)T \)
19 \( 1 + (0.603 - 0.797i)T \)
23 \( 1 + (0.962 - 0.271i)T \)
29 \( 1 + (0.0588 + 0.998i)T \)
31 \( 1 + (0.156 - 0.987i)T \)
37 \( 1 + (-0.0196 - 0.999i)T \)
41 \( 1 + (0.619 - 0.785i)T \)
43 \( 1 + (0.0980 + 0.995i)T \)
47 \( 1 + (-0.852 + 0.522i)T \)
53 \( 1 + (0.252 + 0.967i)T \)
59 \( 1 + (-0.720 - 0.693i)T \)
61 \( 1 + (0.916 + 0.400i)T \)
67 \( 1 + (-0.862 + 0.505i)T \)
71 \( 1 + (0.908 + 0.418i)T \)
73 \( 1 + (0.488 - 0.872i)T \)
79 \( 1 + (0.972 + 0.233i)T \)
83 \( 1 + (0.603 - 0.797i)T \)
89 \( 1 + (0.785 - 0.619i)T \)
97 \( 1 + (-0.156 - 0.987i)T \)
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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

−17.7075455887407263281198184592, −16.83990717232895204844747893383, −16.21349013139543185153597503361, −15.56433448670165856113165409720, −14.91348883605287632606270858688, −13.84118818765064730023309043291, −13.5862273547988878148244840075, −12.95337440204762930080131113720, −12.07029561335173291939128706132, −11.54315388386478357744552150842, −10.9243764671897499919708096974, −10.564498516087559389119909209960, −9.50339019803111774759021149530, −8.38615788437899507968064916490, −8.26044081201763154261003088358, −7.46032779690806702256130615911, −6.643790094444460870946155626270, −6.02510304832800555513468389697, −5.19558965293802209346126222356, −4.93425017503065206542509865976, −3.80333287231989433652109969180, −2.85715508044158076057706767515, −2.07673194337899258987246746240, −1.284518312492097600820625543380, −0.65720426415869065519898032410, 0.445857166132558716811037908825, 1.21542295288979334234608587020, 2.20086029516315542936178190256, 3.079959344096168274002085747790, 4.02898721669484346654429965821, 4.61713507350425713144732504467, 5.11920472323686380288125243325, 5.843207537263956903718345704806, 6.6692754988207849950441115792, 7.358968148807880112019406161128, 8.15654059250392057685132868298, 9.01219977840797891302148949434, 9.37584152767111755078785789866, 10.47889345680330428203432382766, 10.94718620432778582037863529225, 11.20214025535956029453699275475, 12.05448296678508221040887682799, 12.906641007258272161698571939919, 13.418628293080445476778406774797, 14.36771922876438054591590134952, 14.925818168065671826305702532662, 15.62964180826848330158155919830, 15.98070785810766875258125682960, 16.80908429958289313882699720142, 17.5182859003706622584692103947

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