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Properties

Label 1-637-637.446-r1-0-0
Degree $1$
Conductor $637$
Sign $-0.357 + 0.933i$
Analytic cond. $68.4551$
Root an. cond. $68.4551$
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  + (0.733 + 0.680i)2-s + (−0.623 + 0.781i)3-s + (0.0747 + 0.997i)4-s + (−0.988 − 0.149i)5-s + (−0.988 + 0.149i)6-s + (−0.623 + 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.623 − 0.781i)10-s + (0.222 − 0.974i)11-s + (−0.826 − 0.563i)12-s + (0.733 − 0.680i)15-s + (−0.988 + 0.149i)16-s + (−0.826 − 0.563i)17-s + (0.5 − 0.866i)18-s + 19-s + (0.0747 − 0.997i)20-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (−0.623 + 0.781i)3-s + (0.0747 + 0.997i)4-s + (−0.988 − 0.149i)5-s + (−0.988 + 0.149i)6-s + (−0.623 + 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.623 − 0.781i)10-s + (0.222 − 0.974i)11-s + (−0.826 − 0.563i)12-s + (0.733 − 0.680i)15-s + (−0.988 + 0.149i)16-s + (−0.826 − 0.563i)17-s + (0.5 − 0.866i)18-s + 19-s + (0.0747 − 0.997i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.357 + 0.933i$
Analytic conductor: \(68.4551\)
Root analytic conductor: \(68.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (446, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 637,\ (1:\ ),\ -0.357 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9467819426 + 1.376001986i\)
\(L(\frac12)\) \(\approx\) \(0.9467819426 + 1.376001986i\)
\(L(1)\) \(\approx\) \(0.8701651119 + 0.6090422099i\)
\(L(1)\) \(\approx\) \(0.8701651119 + 0.6090422099i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.733 + 0.680i)T \)
3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.0747 + 0.997i)T \)
41 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (-0.988 + 0.149i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (0.365 - 0.930i)T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.826 + 0.563i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−22.5377036208941880435114231051, −21.985317317380788564504086160369, −20.72500583550091322511506850537, −19.81242311324133845311779272369, −19.444961112897928688525036726968, −18.46256803449798302737562530200, −17.78374778958281467927673406503, −16.65406096603000324998645521076, −15.52770030915667074308315122869, −14.98746139374925703593205435487, −13.82713247844016525948870248485, −13.04387279705080153487669312540, −12.23466419988217223647684128182, −11.66182150481564303694890389438, −10.973138557861413121123002449214, −10.03022250186959136038868608542, −8.75840406551091010524276244913, −7.42172902160045119608370114110, −6.88619544103620385826491151865, −5.76761605301439712467641481353, −4.78134109827145488265181005685, −3.96634130358591527181248815668, −2.735658573605936591190389373710, −1.66451670694772003100441435410, −0.55308273286406816274311162907, 0.680037429252727377895312321661, 3.06779744595704501853419481858, 3.618799181369523541247620895792, 4.76318446568835264784378604306, 5.207868602369134638453840938001, 6.48216425665261693554526247267, 7.13027319445659439374011837725, 8.478261110492833682472284275747, 8.95596291531343738683701502369, 10.47309811332108833323926021563, 11.51567265366285233409558322252, 11.79230791260373078471564189821, 12.89725842187789661326364798290, 13.94098407490119569332259802913, 14.83921032607448749002120051048, 15.63107538855984597531413121570, 16.204354380366430402372768314433, 16.74319821566651107868191894885, 17.760916476759801820538787261277, 18.70192005026097612842234101537, 20.069491404277627662739994808573, 20.5942853393257000682873014926, 21.73579577952998614919642460585, 22.1781601204994910895801956045, 23.00959124767324763817404498305

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