Properties

Degree $1$
Conductor $137$
Sign $0.363 + 0.931i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.982 + 0.183i)2-s + (0.602 + 0.798i)3-s + (0.932 − 0.361i)4-s + (0.982 + 0.183i)5-s + (−0.739 − 0.673i)6-s + (0.445 + 0.895i)7-s + (−0.850 + 0.526i)8-s + (−0.273 + 0.961i)9-s − 10-s + (0.932 − 0.361i)11-s + (0.850 + 0.526i)12-s + (−0.445 − 0.895i)13-s + (−0.602 − 0.798i)14-s + (0.445 + 0.895i)15-s + (0.739 − 0.673i)16-s + (−0.850 − 0.526i)17-s + ⋯
L(s,χ)  = 1  + (−0.982 + 0.183i)2-s + (0.602 + 0.798i)3-s + (0.932 − 0.361i)4-s + (0.982 + 0.183i)5-s + (−0.739 − 0.673i)6-s + (0.445 + 0.895i)7-s + (−0.850 + 0.526i)8-s + (−0.273 + 0.961i)9-s − 10-s + (0.932 − 0.361i)11-s + (0.850 + 0.526i)12-s + (−0.445 − 0.895i)13-s + (−0.602 − 0.798i)14-s + (0.445 + 0.895i)15-s + (0.739 − 0.673i)16-s + (−0.850 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.363 + 0.931i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.363 + 0.931i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.363 + 0.931i$
Motivic weight: \(0\)
Character: $\chi_{137} (4, \cdot )$
Sato-Tate group: $\mu(34)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.363 + 0.931i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.8530817817 + 0.5829317756i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.8530817817 + 0.5829317756i\)
\(L(\chi,1)\) \(\approx\) \(0.9129807056 + 0.3824361909i\)
\(L(1,\chi)\) \(\approx\) \(0.9129807056 + 0.3824361909i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.56395123476187938693894720434, −27.06038772424245685838688032500, −26.335196599038698682962899953122, −25.38767040607425741279422843755, −24.60194680161466449082570772331, −23.84816509920218430380669038363, −22.06095577546002577297477356676, −20.80833271778710924097702842416, −20.18108458561909148868572274236, −19.23181172883013247469837876136, −18.12466674826338577335725354265, −17.32825798169335313170412006919, −16.63273438774564217974382754885, −14.74239731656090142014086999835, −13.96639510736960414609332534041, −12.710073567088236559798216089642, −11.63972339284428160710043203602, −10.21567697780508463080839001596, −9.278738134886017197443800781671, −8.28902542415617666208327509833, −7.0567766533633933701726552553, −6.28917252209203935173082285854, −3.98519693049822932297400427821, −2.13913079646047860993490623682, −1.43055733378615693126891469981, 1.94871145350855458955808406839, 2.950831669397772519928884329442, 5.08580307388163279847370229949, 6.17037638677462464177362224838, 7.73376499172391980567343834359, 9.06337812552639145745389573748, 9.38563268738839072361303843775, 10.6592483910344099997493675796, 11.63808399730348150846607566922, 13.49778823186486306650954462763, 14.75082543871904414826085512118, 15.32926159596068869125312358635, 16.59041705699329007682275653029, 17.58275826337175465833441693752, 18.42837769051974473846980209253, 19.742707689644691278633320459870, 20.443210810873361513452126739729, 21.72798659873203329051560879607, 22.12947833501383177956395487976, 24.39698446606008105887648667180, 24.97363872347975724510869736746, 25.75486848079873362494813417500, 26.70745671963029788886513499617, 27.640271339917484891433616334023, 28.28816607246403528889143442756

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