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Properties

Label 1-37-37.4-r0-0-0
Degree $1$
Conductor $37$
Sign $0.568 + 0.822i$
Analytic cond. $0.171827$
Root an. cond. $0.171827$
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  + (0.939 + 0.342i)2-s + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s − 6-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.939 − 0.342i)12-s + (−0.766 − 0.642i)13-s + (0.5 − 0.866i)14-s + (−0.173 − 0.984i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s − 6-s + (0.173 − 0.984i)7-s + (0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.939 − 0.342i)12-s + (−0.766 − 0.642i)13-s + (0.5 − 0.866i)14-s + (−0.173 − 0.984i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(0.171827\)
Root analytic conductor: \(0.171827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (0:\ ),\ 0.568 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8787372549 + 0.4609045959i\)
\(L(\frac12)\) \(\approx\) \(0.8787372549 + 0.4609045959i\)
\(L(1)\) \(\approx\) \(1.123598560 + 0.4089115837i\)
\(L(1)\) \(\approx\) \(1.123598560 + 0.4089115837i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
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

−35.25935615800503360856869541956, −34.00578011127509228956629443688, −33.106563757499984006956352327443, −31.573185365058292407100605050915, −30.92262042465504229963397300848, −29.11222522549157177830835037960, −28.679421662204383218571231541953, −27.55906517346149933677361928642, −24.99386624508049543001698290375, −24.29879934126055003253141698307, −23.17401530952307696889276063649, −22.01318600425391518407037624594, −20.916164729151314949238860838387, −19.48281641865946430642242282296, −17.97359029256949552537345940906, −16.3557325439038188784050319832, −15.30165804662761050999996213147, −13.35399475243116855563969228195, −12.23047145217656675123557078187, −11.57373458813657268850334328110, −9.638216025683352254015712285, −7.277734240068670771393366488251, −5.501891530333362684809077623007, −4.68474676720624471414784284610, −1.96857853701371214787576443373, 3.291691085377348818839383657, 4.85286808562406388641169668955, 6.383182436488355326640329945177, 7.53876337199953804478046197912, 10.53716313896090848901234463595, 11.228561564707156485106023410437, 12.877992775421300087163502437316, 14.34087446506882071022690815016, 15.579904108370806517940057977361, 16.78083004521403710349852557628, 18.00784423375032171794680210689, 20.00733320929104349784169392858, 21.58055241285770017388429251539, 22.409853239583000512791445170574, 23.40743942802893249467568705314, 24.36206537637199701703469389449, 26.34920792999732410862567890408, 26.98124076013339523776655747498, 29.07012487597618402823945668348, 29.82167505940103525155116089043, 30.95705419702737356231933349234, 32.58786169063507319100818184731, 33.3428788792064425021617968129, 34.48632407436252154728660299180, 35.03640964536295017950215913062

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