Properties

Label 1-137-137.69-r0-0-0
Degree $1$
Conductor $137$
Sign $0.822 + 0.568i$
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 + (0.895 − 0.445i)3-s + (−0.982 − 0.183i)4-s + (−0.995 + 0.0922i)5-s + (0.361 + 0.932i)6-s + (0.850 − 0.526i)7-s + (0.273 − 0.961i)8-s + (0.602 − 0.798i)9-s i·10-s + (0.982 + 0.183i)11-s + (−0.961 + 0.273i)12-s + (0.526 + 0.850i)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 + (0.895 − 0.445i)3-s + (−0.982 − 0.183i)4-s + (−0.995 + 0.0922i)5-s + (0.361 + 0.932i)6-s + (0.850 − 0.526i)7-s + (0.273 − 0.961i)8-s + (0.602 − 0.798i)9-s i·10-s + (0.982 + 0.183i)11-s + (−0.961 + 0.273i)12-s + (0.526 + 0.850i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.822 + 0.568i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.822 + 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.164857533 + 0.3633076619i\)
\(L(\frac12)\) \(\approx\) \(1.164857533 + 0.3633076619i\)
\(L(1)\) \(\approx\) \(1.126788706 + 0.3131762656i\)
\(L(1)\) \(\approx\) \(1.126788706 + 0.3131762656i\)

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 \( 1 + (0.895 - 0.445i)T \)
5 \( 1 + (-0.995 + 0.0922i)T \)
7 \( 1 + (0.850 - 0.526i)T \)
11 \( 1 + (0.982 + 0.183i)T \)
13 \( 1 + (0.526 + 0.850i)T \)
17 \( 1 + (0.273 + 0.961i)T \)
19 \( 1 + (-0.739 - 0.673i)T \)
23 \( 1 + (0.361 - 0.932i)T \)
29 \( 1 + (-0.361 + 0.932i)T \)
31 \( 1 + (-0.673 - 0.739i)T \)
37 \( 1 - T \)
41 \( 1 + iT \)
43 \( 1 + (0.673 - 0.739i)T \)
47 \( 1 + (-0.798 - 0.602i)T \)
53 \( 1 + (-0.673 + 0.739i)T \)
59 \( 1 + (-0.602 + 0.798i)T \)
61 \( 1 + (0.602 + 0.798i)T \)
67 \( 1 + (-0.526 - 0.850i)T \)
71 \( 1 + (0.183 + 0.982i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (-0.895 - 0.445i)T \)
83 \( 1 + (0.961 + 0.273i)T \)
89 \( 1 + (0.995 - 0.0922i)T \)
97 \( 1 + (-0.183 + 0.982i)T \)
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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

−27.81438876043882979527229470224, −27.612968591278682572113157107497, −26.9229999491060076750761467547, −25.53317240466677036744170558976, −24.533140850134422317892067003349, −23.147007011178806658395251855736, −22.20125139023681247027116457015, −21.020000036942493776767955084821, −20.48266300515168079810815162644, −19.40819568164129673763590443574, −18.79125982963190391860592135543, −17.47621793578507696190046737177, −15.9890833343784874734736857182, −14.866285595898784823033413587926, −14.0764657386759989330795061232, −12.7199140267120747034468788228, −11.63044739229718632487769106237, −10.78891766393895217698980249614, −9.36356357630802385842004567064, −8.49576720126164633405790943508, −7.69401785927466322016833813934, −5.18051698131768052043054889921, −4.00752747960033668291636257171, −3.139978851045901093646176807158, −1.57249590054886059386718649713, 1.41428169222116444757575866218, 3.76431833484062060665612643370, 4.448774693016682218113241741459, 6.552807609299151567765246402253, 7.31828837938352851704847275695, 8.38906665791094248584748783437, 9.02285296625082744183377917341, 10.78271164894939924530842282797, 12.24947710168374213272743057630, 13.45156192440904896665952752175, 14.71248051104395871593001280880, 14.82479518712804996910275716405, 16.3345415816552901098557470395, 17.34174285254897153237759335083, 18.584096545445258821389325996101, 19.32426998701086442996441399895, 20.32305151911687952312041545849, 21.66210571161857129050110758021, 23.10628415059012877323182777455, 23.9448663627897686590909385948, 24.40390094658774999061536710913, 25.73519139222662952157472697998, 26.402375355268110566567380128996, 27.34200432681821779501982621982, 28.083742044354375172600337069915

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