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Properties

Label 1-308-308.151-r0-0-0
Degree $1$
Conductor $308$
Sign $0.979 - 0.201i$
Analytic cond. $1.43034$
Root an. cond. $1.43034$
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  + (−0.913 − 0.406i)3-s + (−0.978 − 0.207i)5-s + (0.669 + 0.743i)9-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.5 − 0.866i)23-s + (0.913 + 0.406i)25-s + (−0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.978 − 0.207i)31-s + (0.913 − 0.406i)37-s + (0.669 − 0.743i)39-s + (0.809 − 0.587i)41-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)3-s + (−0.978 − 0.207i)5-s + (0.669 + 0.743i)9-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.5 − 0.866i)23-s + (0.913 + 0.406i)25-s + (−0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.978 − 0.207i)31-s + (0.913 − 0.406i)37-s + (0.669 − 0.743i)39-s + (0.809 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.979 - 0.201i$
Analytic conductor: \(1.43034\)
Root analytic conductor: \(1.43034\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (0:\ ),\ 0.979 - 0.201i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7019404250 - 0.07159234050i\)
\(L(\frac12)\) \(\approx\) \(0.7019404250 - 0.07159234050i\)
\(L(1)\) \(\approx\) \(0.6940891563 - 0.06703076216i\)
\(L(1)\) \(\approx\) \(0.6940891563 - 0.06703076216i\)

Euler product

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

−25.16439104776704869180973928397, −24.292797489999685106002656989332, −23.10996453854184919705471722907, −22.920558251784617927678627266360, −21.9065335913029032581933221903, −20.87368760781172740340117880669, −19.94213307856876520502313906993, −18.94085770114810458106440600452, −17.967722284334474669227973418984, −17.16516574246059839610280590389, −16.03491740869517517703021463722, −15.52825556026088323959973321721, −14.59173448120043607649639622202, −13.11987227398459947248980837077, −12.11756155239051653838824856101, −11.43614493479133219545267490451, −10.53107511410757666240893242349, −9.60290605152767159455671773667, −8.1951835997773589246373495048, −7.233186561404093629416936752972, −6.14741534559136232876244864236, −4.9932746242893020965599687560, −4.08668413698928931817069355557, −2.906760946392884221314656075586, −0.79927701165089484643510055987, 0.85425281970303373619390752532, 2.44989518162027997396366995978, 4.22494225884287801383746697353, 4.81811303132114998645067194119, 6.325862580778295191209312413216, 7.0529299181049652546135454839, 8.129013534027811362280509675, 9.227245199795629223831534446868, 10.7449936347862054115488840766, 11.305989117729891540666065072244, 12.30828811199453271749478572032, 12.94217229906536467735919671315, 14.24034280181718174471707950535, 15.46080486172497631591181026016, 16.199729620984549290992115074594, 17.0927882815837622092789349641, 17.92155632258253169296468123941, 19.16130329478366236583521715448, 19.4540916785931484553125171646, 20.81886036246992194677356025277, 21.9127545063529983678404312870, 22.61657104609543661119513107189, 23.73623910083517264543134477162, 23.967258972809887925038621567530, 24.968280547892353844817280226567

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