Properties

Degree $1$
Conductor $311$
Sign $-0.889 - 0.457i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.612 − 0.790i)2-s + (−0.612 + 0.790i)3-s + (−0.250 + 0.968i)4-s + (0.528 − 0.848i)5-s + 6-s + (−0.0506 − 0.998i)7-s + (0.918 − 0.394i)8-s + (−0.250 − 0.968i)9-s + (−0.994 + 0.101i)10-s + (−0.440 + 0.897i)11-s + (−0.612 − 0.790i)12-s + (−0.250 − 0.968i)13-s + (−0.758 + 0.651i)14-s + (0.347 + 0.937i)15-s + (−0.874 − 0.485i)16-s + (−0.994 + 0.101i)17-s + ⋯
L(s,χ)  = 1  + (−0.612 − 0.790i)2-s + (−0.612 + 0.790i)3-s + (−0.250 + 0.968i)4-s + (0.528 − 0.848i)5-s + 6-s + (−0.0506 − 0.998i)7-s + (0.918 − 0.394i)8-s + (−0.250 − 0.968i)9-s + (−0.994 + 0.101i)10-s + (−0.440 + 0.897i)11-s + (−0.612 − 0.790i)12-s + (−0.250 − 0.968i)13-s + (−0.758 + 0.651i)14-s + (0.347 + 0.937i)15-s + (−0.874 − 0.485i)16-s + (−0.994 + 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.889 - 0.457i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.889 - 0.457i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $-0.889 - 0.457i$
Motivic weight: \(0\)
Character: $\chi_{311} (265, \cdot )$
Sato-Tate group: $\mu(31)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ -0.889 - 0.457i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.1044782901 - 0.4317531898i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.1044782901 - 0.4317531898i\)
\(L(\chi,1)\) \(\approx\) \(0.5097788378 - 0.2551073055i\)
\(L(1,\chi)\) \(\approx\) \(0.5097788378 - 0.2551073055i\)

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

−25.5976455766410764661889233250, −24.61189928823559491862299801546, −24.160286075207427355214622618395, −23.13390258081456669049037778647, −22.13304774496178598365957322823, −21.59592088063963909802886816306, −19.68363497727190427130262627914, −18.88509761403416149850045718849, −18.32576353876071701525403300144, −17.70064487747843928397323802533, −16.66931812739661399816928477562, −15.78338023610530868923668576069, −14.73214992967901089530436990174, −13.78785594160485966646210407036, −12.98661657396843897579376989726, −11.27375353306759080216381526130, −11.05005445449578851820154654630, −9.48369289603136253811556898874, −8.71942808322587725644629145815, −7.36677058275260642278476177847, −6.6956532931226881601148951302, −5.81273218032557283525736964174, −5.055433195536207725309131760786, −2.71368783679427789490750304871, −1.65355477695450926081097049256, 0.37429581157510235489641186195, 1.80013117340619483018509092616, 3.42655685232090322440391290, 4.48987120315284388398274964666, 5.286271768287994487566619383450, 6.9135086256482606869727870084, 8.12565118696151685929473812311, 9.278242182861024181186018152414, 10.081484913843157156824872039352, 10.58847437405337467015849656378, 11.75727841229786255773275029168, 12.77979371147055829744655995167, 13.4002527706685967107817385445, 14.99419127717190611072479679419, 16.15863448731751602827137979119, 16.981201633014592378184309958058, 17.46616514639508725724541087020, 18.30020566078391067040035791288, 19.901395149928477735100214871272, 20.57455041641021306799015776470, 20.8291276052620027810135249202, 22.18680943498664138371105996210, 22.70408147603562831426999365778, 23.818637906258081539652858624390, 25.12378847231742902855003860390

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