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Properties

Label 1-308-308.247-r1-0-0
Degree $1$
Conductor $308$
Sign $-0.515 + 0.857i$
Analytic cond. $33.0991$
Root an. cond. $33.0991$
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  + (0.978 + 0.207i)3-s + (−0.104 + 0.994i)5-s + (0.913 + 0.406i)9-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.913 − 0.406i)17-s + (−0.669 + 0.743i)19-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (0.104 + 0.994i)31-s + (−0.978 + 0.207i)37-s + (−0.913 + 0.406i)39-s + (0.309 + 0.951i)41-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)3-s + (−0.104 + 0.994i)5-s + (0.913 + 0.406i)9-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.913 − 0.406i)17-s + (−0.669 + 0.743i)19-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (0.104 + 0.994i)31-s + (−0.978 + 0.207i)37-s + (−0.913 + 0.406i)39-s + (0.309 + 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.515 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.515 + 0.857i$
Analytic conductor: \(33.0991\)
Root analytic conductor: \(33.0991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (1:\ ),\ -0.515 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.111867934 + 1.965592048i\)
\(L(\frac12)\) \(\approx\) \(1.111867934 + 1.965592048i\)
\(L(1)\) \(\approx\) \(1.271462730 + 0.5701642483i\)
\(L(1)\) \(\approx\) \(1.271462730 + 0.5701642483i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.669 + 0.743i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−24.719936351953027391727193916532, −24.13400506233969488136657709789, −23.18150169366420213299090864184, −21.814296483477987370859244540497, −21.00056935931262471499397062511, −20.19899992833747451625360545277, −19.52685440245760298293266828745, −18.65781114417052326010611451932, −17.41713470352434358474162638395, −16.59853439627851199507201316436, −15.441662507393403424220461788161, −14.73853750052941569955869542190, −13.65419923754245868460345842783, −12.709877607957238446229956940512, −12.2174148934627880723412768001, −10.56831935933718902435680646860, −9.54072985099581689504810814299, −8.65106393152665220820065996851, −7.914332508759025543192675387, −6.84513380699935999660093135051, −5.32150175334485237430329675006, −4.31795268793421520023597161568, −3.10490422519015377487135480621, −1.88113918805251901108734332674, −0.56616628299777447419546540748, 1.72379210750462574741401978931, 2.86219398708436823365691758639, 3.69832596327307545082318567699, 4.957273819898270862061502416635, 6.52455190783750830698429118508, 7.4343879987341080981919248724, 8.28047588736969400456913928960, 9.62194372169962286301351500842, 10.15601592197967153643763339044, 11.384205611960105701793486833593, 12.470554783508271362715078563922, 13.74292138973079424015128549305, 14.413163049563308008905439005963, 15.0858192468093798805230960363, 16.05823056869951337977687722349, 17.20339901166325093294460711404, 18.43504250098917654690631372273, 19.16099243405253162148505235473, 19.72949470335181237347115146807, 21.11538015248869487220461754330, 21.49360867763465949062003508614, 22.67142899454233070490277884711, 23.50007498138555730274957797900, 24.71780760537354272554429509561, 25.451121808054546345927067893259

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