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Properties

Label 1-2600-2600.1589-r0-0-0
Degree $1$
Conductor $2600$
Sign $-0.983 - 0.183i$
Analytic cond. $12.0743$
Root an. cond. $12.0743$
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  + (0.913 − 0.406i)3-s + (0.5 + 0.866i)7-s + (0.669 − 0.743i)9-s + (−0.669 − 0.743i)11-s + (−0.913 − 0.406i)17-s + (−0.913 − 0.406i)19-s + (0.809 + 0.587i)21-s + (−0.669 − 0.743i)23-s + (0.309 − 0.951i)27-s + (−0.913 + 0.406i)29-s + (−0.809 + 0.587i)31-s + (−0.913 − 0.406i)33-s + (−0.978 − 0.207i)37-s + (−0.978 − 0.207i)41-s + (−0.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (0.5 + 0.866i)7-s + (0.669 − 0.743i)9-s + (−0.669 − 0.743i)11-s + (−0.913 − 0.406i)17-s + (−0.913 − 0.406i)19-s + (0.809 + 0.587i)21-s + (−0.669 − 0.743i)23-s + (0.309 − 0.951i)27-s + (−0.913 + 0.406i)29-s + (−0.809 + 0.587i)31-s + (−0.913 − 0.406i)33-s + (−0.978 − 0.207i)37-s + (−0.978 − 0.207i)41-s + (−0.5 − 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $-0.983 - 0.183i$
Analytic conductor: \(12.0743\)
Root analytic conductor: \(12.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (0:\ ),\ -0.983 - 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05950534399 - 0.6434400491i\)
\(L(\frac12)\) \(\approx\) \(0.05950534399 - 0.6434400491i\)
\(L(1)\) \(\approx\) \(1.076058986 - 0.2331139842i\)
\(L(1)\) \(\approx\) \(1.076058986 - 0.2331139842i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−19.978487147480306328252821920063, −19.067253795423161206497022219407, −18.38716077666984872066015956622, −17.484746809823857652970616616101, −16.939905657248636900137296600169, −16.01637716800130242704518517339, −15.282425411326493364480298472133, −14.82176938675935881280385422539, −14.05305805023244995629383750356, −13.25180657094059819617314325146, −12.929175271808963351604742266797, −11.726651945069324204941057853131, −10.808644315473615562517763127016, −10.31556666909306429068373459107, −9.62276585575227040539192249073, −8.75091346827425053433854909467, −7.97901866292876125612331434232, −7.4953283433963508734019577653, −6.6653976768396881725689551203, −5.48368904380340650648855477053, −4.56282887505409340363499847702, −4.068075096821288115867170333745, −3.27442091047414589258134582619, −2.01024170395110679845618106408, −1.75407057334481763162824567018, 0.15032411163507494896846011082, 1.68624509224589096886627027371, 2.26863372871681515580549466441, 3.01404377091183383880001090586, 3.93523726960436479868864256430, 4.89714347393634728353130572116, 5.7160339804293634311497181759, 6.642884266637817437221176941607, 7.348634004739854600494679609059, 8.39000983687879655542873309334, 8.62055309701758353349602489523, 9.313701508380812475042492292158, 10.4192223009898048555232987820, 11.087036675103091941380900586950, 11.987785272985055657099057467712, 12.728896256134704409777960137698, 13.30127434017981547960333978554, 14.13176462177585193255461784413, 14.65901718999568214598224140994, 15.563194651008144275159536867712, 15.83193396853872684121436126604, 17.00665790403910619153028148723, 17.86895391244255863861461716006, 18.53369752113300322693166860620, 18.84609189919503321969604817726

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