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Properties

Label 1-137-137.14-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.904 - 0.426i$
Analytic cond. $0.636225$
Root an. cond. $0.636225$
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  + (0.445 − 0.895i)2-s + (−0.739 + 0.673i)3-s + (−0.602 − 0.798i)4-s + (−0.445 − 0.895i)5-s + (0.273 + 0.961i)6-s + (0.932 + 0.361i)7-s + (−0.982 + 0.183i)8-s + (0.0922 − 0.995i)9-s − 10-s + (−0.602 − 0.798i)11-s + (0.982 + 0.183i)12-s + (−0.932 − 0.361i)13-s + (0.739 − 0.673i)14-s + (0.932 + 0.361i)15-s + (−0.273 + 0.961i)16-s + (−0.982 − 0.183i)17-s + ⋯
L(s)  = 1  + (0.445 − 0.895i)2-s + (−0.739 + 0.673i)3-s + (−0.602 − 0.798i)4-s + (−0.445 − 0.895i)5-s + (0.273 + 0.961i)6-s + (0.932 + 0.361i)7-s + (−0.982 + 0.183i)8-s + (0.0922 − 0.995i)9-s − 10-s + (−0.602 − 0.798i)11-s + (0.982 + 0.183i)12-s + (−0.932 − 0.361i)13-s + (0.739 − 0.673i)14-s + (0.932 + 0.361i)15-s + (−0.273 + 0.961i)16-s + (−0.982 − 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.904 - 0.426i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.904 - 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1513586974 - 0.6753640846i\)
\(L(\frac12)\) \(\approx\) \(0.1513586974 - 0.6753640846i\)
\(L(1)\) \(\approx\) \(0.6369017687 - 0.5001221540i\)
\(L(1)\) \(\approx\) \(0.6369017687 - 0.5001221540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.445 - 0.895i)T \)
3 \( 1 + (-0.739 + 0.673i)T \)
5 \( 1 + (-0.445 - 0.895i)T \)
7 \( 1 + (0.932 + 0.361i)T \)
11 \( 1 + (-0.602 - 0.798i)T \)
13 \( 1 + (-0.932 - 0.361i)T \)
17 \( 1 + (-0.982 - 0.183i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (0.273 - 0.961i)T \)
29 \( 1 + (0.273 - 0.961i)T \)
31 \( 1 + (0.850 - 0.526i)T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + (0.850 + 0.526i)T \)
47 \( 1 + (-0.0922 + 0.995i)T \)
53 \( 1 + (0.850 + 0.526i)T \)
59 \( 1 + (0.0922 - 0.995i)T \)
61 \( 1 + (0.0922 + 0.995i)T \)
67 \( 1 + (-0.932 - 0.361i)T \)
71 \( 1 + (0.602 - 0.798i)T \)
73 \( 1 + (0.932 - 0.361i)T \)
79 \( 1 + (-0.739 - 0.673i)T \)
83 \( 1 + (0.982 - 0.183i)T \)
89 \( 1 + (-0.445 - 0.895i)T \)
97 \( 1 + (0.602 + 0.798i)T \)
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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

−29.10993191522478339097341276728, −27.598551113494290460722870996099, −26.932016442513727856874850756941, −25.80311352667810361476304369539, −24.73931687104254504729395338747, −23.67120953198372941706347054732, −23.336570389707936840908592459585, −22.242987772354623163063183854600, −21.38273913510089021747900580543, −19.67842803589765341739169089996, −18.38092768770199368299514126825, −17.698725089192260833017816073099, −16.900785406380537235922655441329, −15.49682255022680099472907512220, −14.663014373621312690111976465016, −13.61624631782712055349332876556, −12.431267603777397155479652815686, −11.48153728971466701972950209840, −10.31444008345737643818442244142, −8.268228280339021584886217378358, −7.309839933676483029519705955745, −6.7246728660476574503606664209, −5.22623961443522521695501853983, −4.26351387102396963698181738345, −2.28734230030178440858642385822, 0.57677832911037114199079512212, 2.56041308522518871979001300001, 4.41170226522358660109056270833, 4.829226543072681806589010944660, 6.01193194099715327608082793358, 8.26758709888365899789911528398, 9.27723013960039803657988648169, 10.62694251174617650184847559682, 11.41864023323878065296540314817, 12.26759318562946933230449149618, 13.31241823536522809969769698582, 14.87586194423071207589839409853, 15.60355333778649379099858271737, 16.99946781459529550703228480302, 17.91084830766463633480821793842, 19.19010502503936210277485562777, 20.38431404348535445610002777071, 21.12001327849986418100023226855, 21.863325805848936283282943053210, 22.92506559017504025374303406382, 24.031862042009813243993124393261, 24.45812266650645506746897398965, 26.81163943561112619248431383570, 27.23049146513384562276326063393, 28.315928433349219724520514235157

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