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Properties

Label 1-137-137.65-r0-0-0
Degree $1$
Conductor $137$
Sign $0.924 - 0.380i$
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  + (−0.602 + 0.798i)2-s + (−0.0922 − 0.995i)3-s + (−0.273 − 0.961i)4-s + (0.602 + 0.798i)5-s + (0.850 + 0.526i)6-s + (0.739 − 0.673i)7-s + (0.932 + 0.361i)8-s + (−0.982 + 0.183i)9-s − 10-s + (−0.273 − 0.961i)11-s + (−0.932 + 0.361i)12-s + (−0.739 + 0.673i)13-s + (0.0922 + 0.995i)14-s + (0.739 − 0.673i)15-s + (−0.850 + 0.526i)16-s + (0.932 − 0.361i)17-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (−0.0922 − 0.995i)3-s + (−0.273 − 0.961i)4-s + (0.602 + 0.798i)5-s + (0.850 + 0.526i)6-s + (0.739 − 0.673i)7-s + (0.932 + 0.361i)8-s + (−0.982 + 0.183i)9-s − 10-s + (−0.273 − 0.961i)11-s + (−0.932 + 0.361i)12-s + (−0.739 + 0.673i)13-s + (0.0922 + 0.995i)14-s + (0.739 − 0.673i)15-s + (−0.850 + 0.526i)16-s + (0.932 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.924 - 0.380i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.924 - 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8377079336 - 0.1656410573i\)
\(L(\frac12)\) \(\approx\) \(0.8377079336 - 0.1656410573i\)
\(L(1)\) \(\approx\) \(0.8518550107 + 0.02897209245i\)
\(L(1)\) \(\approx\) \(0.8518550107 + 0.02897209245i\)

Euler product

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

−28.548640145642230154042857365795, −27.4849766194518360420459805566, −27.15354796558543951510775943402, −25.47414439975629312092197912358, −25.19386112616916126632099214241, −23.338695080898922589010016142281, −22.06943270341923763856458043953, −21.335431518804028364331663550083, −20.635798948314638764759146433856, −19.87223619593444347138789959670, −18.277332135080281078266296336438, −17.40915380666694986271140713722, −16.682600854703543813395672354826, −15.396261571416432687524001734454, −14.21985726025204609625135105531, −12.58665015768194343466269100221, −11.957880323331367738330910254468, −10.48902633193292523640704324855, −9.78124054821950573411442393961, −8.82472253502230465766536842854, −7.79850836170390753988779739228, −5.4142991811790550036416099532, −4.69454580427861723767204877631, −3.06090886294798721737990609961, −1.631654879415608627432534476685, 1.08476838623595710990175851320, 2.613112371411454498271695432459, 5.02743995471339834938809139378, 6.207090990911960122624805451289, 7.17562437860668128124575763269, 7.9290899856163770217810751840, 9.31264556798080456689798267146, 10.65748669660996566936952609235, 11.52852243741932499351901298272, 13.48776795144110724965805173712, 14.044789340310148591909297434849, 14.90869118614249992497113797379, 16.68678203257458973108140606111, 17.26791147383086448786473165927, 18.36313941513117977772907096138, 18.86796159033749903166419265460, 20.01987260140748490769202252058, 21.551658726142632880277666144259, 22.86434030521871795044768988778, 23.75657842313734911295504396642, 24.492370137851335567601189212057, 25.39051793085414726155015051940, 26.47477361694011347370681898081, 27.02200201603585334368243754287, 28.55891719554044266931388643858

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