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Properties

Label 1-311-311.106-r0-0-0
Degree $1$
Conductor $311$
Sign $0.338 + 0.941i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
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  + (−0.289 + 0.957i)2-s + (−0.328 + 0.944i)3-s + (−0.832 − 0.554i)4-s + (0.458 − 0.888i)5-s + (−0.809 − 0.587i)6-s + (0.528 + 0.848i)7-s + (0.771 − 0.635i)8-s + (−0.784 − 0.620i)9-s + (0.717 + 0.696i)10-s + (−0.954 − 0.299i)11-s + (0.796 − 0.604i)12-s + (0.347 − 0.937i)13-s + (−0.965 + 0.260i)14-s + (0.688 + 0.724i)15-s + (0.385 + 0.922i)16-s + (0.884 − 0.467i)17-s + ⋯
L(s)  = 1  + (−0.289 + 0.957i)2-s + (−0.328 + 0.944i)3-s + (−0.832 − 0.554i)4-s + (0.458 − 0.888i)5-s + (−0.809 − 0.587i)6-s + (0.528 + 0.848i)7-s + (0.771 − 0.635i)8-s + (−0.784 − 0.620i)9-s + (0.717 + 0.696i)10-s + (−0.954 − 0.299i)11-s + (0.796 − 0.604i)12-s + (0.347 − 0.937i)13-s + (−0.965 + 0.260i)14-s + (0.688 + 0.724i)15-s + (0.385 + 0.922i)16-s + (0.884 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.338 + 0.941i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.338 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8144319790 + 0.5727617537i\)
\(L(\frac12)\) \(\approx\) \(0.8144319790 + 0.5727617537i\)
\(L(1)\) \(\approx\) \(0.7690737130 + 0.4402334726i\)
\(L(1)\) \(\approx\) \(0.7690737130 + 0.4402334726i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (-0.289 + 0.957i)T \)
3 \( 1 + (-0.328 + 0.944i)T \)
5 \( 1 + (0.458 - 0.888i)T \)
7 \( 1 + (0.528 + 0.848i)T \)
11 \( 1 + (-0.954 - 0.299i)T \)
13 \( 1 + (0.347 - 0.937i)T \)
17 \( 1 + (0.884 - 0.467i)T \)
19 \( 1 + (0.986 - 0.161i)T \)
23 \( 1 + (0.843 + 0.537i)T \)
29 \( 1 + (-0.546 + 0.837i)T \)
31 \( 1 + (0.717 - 0.696i)T \)
37 \( 1 + (0.0708 + 0.997i)T \)
41 \( 1 + (0.979 + 0.201i)T \)
43 \( 1 + (0.970 - 0.240i)T \)
47 \( 1 + (-0.0506 - 0.998i)T \)
53 \( 1 + (-0.926 - 0.375i)T \)
59 \( 1 + (-0.643 + 0.765i)T \)
61 \( 1 + (0.151 - 0.988i)T \)
67 \( 1 + (-0.674 + 0.738i)T \)
71 \( 1 + (-0.403 + 0.914i)T \)
73 \( 1 + (-0.731 + 0.681i)T \)
79 \( 1 + (0.111 + 0.993i)T \)
83 \( 1 + (0.688 - 0.724i)T \)
89 \( 1 + (0.528 - 0.848i)T \)
97 \( 1 + (0.745 + 0.666i)T \)
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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

−25.20755554682956164049020865073, −23.902393197653368013957910462522, −23.10417107387798076134942924145, −22.55504805680102997600671382396, −21.19689982793490092415796295479, −20.750769319104305008011421141544, −19.36206700235398312171198553802, −18.82357375608539427842157147889, −17.95108164136798664380432422342, −17.422151196945802004650427839658, −16.38890775093782252114441202591, −14.45088721777899390261505882163, −13.88563209657133393009880919178, −13.05915887541226770861968588742, −12.01961324244816631232984188587, −10.98938290370035146999303100137, −10.55155262579884460900433530309, −9.33421567382488420899390966440, −7.8220071533393505855255609185, −7.37823986900095317211116630225, −5.98113158263477474623445139377, −4.70157677569168949914141408039, −3.23753203619708657798091084759, −2.16060985789099677615881582361, −1.14467510198775898537519922338, 0.97984110801698350304864036258, 3.07396145697544387981710399653, 4.731057509947049146152922876823, 5.43370925939260961410675418027, 5.8085766362518476605294716273, 7.69428089885152535068380973800, 8.55899589241008922800577789488, 9.3815692162975350344472992161, 10.18190009659178764381702017274, 11.37088533734365222912756026666, 12.66636349762251210251686782456, 13.68961994985885110527971608178, 14.7890625668485063883530485056, 15.67618417727946367192895689732, 16.15758686975133325597180552616, 17.18850093200478929219339834675, 17.89782328807204262958620888356, 18.73550799890074094868068568534, 20.31993189790378618775371941644, 21.00758890851530952775818068785, 21.89159952117601121071847886289, 22.8338974309797279638882478639, 23.75205206224552766226082077177, 24.647441352086615572399321473, 25.406833820370587793276439365858

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