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Properties

Label 1-137-137.63-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.443 - 0.896i$
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  + (0.0922 + 0.995i)2-s + (−0.445 + 0.895i)3-s + (−0.982 + 0.183i)4-s + (−0.0922 + 0.995i)5-s + (−0.932 − 0.361i)6-s + (−0.850 − 0.526i)7-s + (−0.273 − 0.961i)8-s + (−0.602 − 0.798i)9-s − 10-s + (−0.982 + 0.183i)11-s + (0.273 − 0.961i)12-s + (0.850 + 0.526i)13-s + (0.445 − 0.895i)14-s + (−0.850 − 0.526i)15-s + (0.932 − 0.361i)16-s + (−0.273 + 0.961i)17-s + ⋯
L(s)  = 1  + (0.0922 + 0.995i)2-s + (−0.445 + 0.895i)3-s + (−0.982 + 0.183i)4-s + (−0.0922 + 0.995i)5-s + (−0.932 − 0.361i)6-s + (−0.850 − 0.526i)7-s + (−0.273 − 0.961i)8-s + (−0.602 − 0.798i)9-s − 10-s + (−0.982 + 0.183i)11-s + (0.273 − 0.961i)12-s + (0.850 + 0.526i)13-s + (0.445 − 0.895i)14-s + (−0.850 − 0.526i)15-s + (0.932 − 0.361i)16-s + (−0.273 + 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.443 - 0.896i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.443 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2198762631 + 0.3541780564i\)
\(L(\frac12)\) \(\approx\) \(-0.2198762631 + 0.3541780564i\)
\(L(1)\) \(\approx\) \(0.3192517654 + 0.5298580482i\)
\(L(1)\) \(\approx\) \(0.3192517654 + 0.5298580482i\)

Euler product

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

−28.44282326342158020860282814970, −27.21753060314771831200522363795, −25.73644491658876217055074796255, −24.69602918554496978101667547844, −23.6049073422091393458856343959, −22.86549561490444841004101499946, −21.8711775712839107065992651391, −20.51254708879675958233949881839, −19.94008352451781516930232329111, −18.46870098879441897120680260309, −18.32296815046839282063082303125, −16.7419939372223785223123437198, −15.76625434856047301636241926952, −13.75219596688325459064690128932, −13.07972414487985316551163246210, −12.334482071071639477257753574962, −11.43294034006222630322797303746, −10.09186437728329785671393985391, −8.837220920999282501683224086956, −7.843383191068729533130209772795, −5.89879243264698772521509568893, −5.139646614536655374753259524278, −3.33745893417579373563477337511, −1.94237013184351644571041903586, −0.373866915045249516377633126998, 3.294598539771841246267162156322, 4.18608732160242338311245910507, 5.73327352392182472542415209998, 6.553371180313267602216033550635, 7.75013193464827650801344219399, 9.314662855975456747209848809554, 10.21365456886769876576836684111, 11.234709181500062716021773257525, 12.97130000186059820639773289315, 14.0038242324281586952796019989, 15.20120787924122423287001634370, 15.82922407249259114827818695529, 16.72347818951863227995176479633, 17.88006819372668833724201354405, 18.7194809789375009987514134048, 20.20719426821675404856169350027, 21.73685415725265991422142982829, 22.234051318514608325728861261953, 23.34379406581495634728987547753, 23.74419516894603360433388841842, 25.82116882072755125428280570353, 26.0759481303326030195731796922, 26.75949947150620165967990460032, 28.01442977019636438197223714896, 28.88564263004997127059111430993

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