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Properties

Label 1-308-308.263-r0-0-0
Degree $1$
Conductor $308$
Sign $-0.832 + 0.553i$
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  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s − 13-s − 15-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)39-s − 41-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s − 13-s − 15-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(1.43034\)
Root analytic conductor: \(1.43034\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 308,\ (0:\ ),\ -0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2958595638 + 0.9789728682i\)
\(L(\frac12)\) \(\approx\) \(0.2958595638 + 0.9789728682i\)
\(L(1)\) \(\approx\) \(0.8216720009 + 0.5613795103i\)
\(L(1)\) \(\approx\) \(0.8216720009 + 0.5613795103i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−24.79809040617108355312214759900, −24.10137295541518355085017849722, −23.43241604711022303627377200570, −22.36752008387141593246410571843, −21.02991799298758321036468457474, −20.33107353057133896350470348495, −19.432982083688922491083339765658, −18.89740150384593029984460643078, −17.57337177062452296257301268862, −16.94761531307060188527544607264, −15.69044169271242919704419512645, −14.825849659761913685344781616630, −13.72897152330874145565907010536, −12.89481764124528566806548773511, −12.116904831523714171686209005668, −11.28626683648588818190907001228, −9.54936349009798723326306097832, −8.91070342428511292319246408149, −7.70141990636968618281480319255, −7.195563471949617867144977455265, −5.69755545232796310167198233988, −4.58676202877682710130315998296, −3.24486972461130637943864318324, −2.02381445982983087970730851184, −0.60427306270246958936823447881, 2.1876517611262941031620154897, 3.27765183791849971007340268789, 4.14195880252262852324553752309, 5.33332628698215206832456148801, 6.70161705065263241361147825177, 7.82685205304386973413277430372, 8.66356016147433476742688318282, 10.049853701704169583524105694179, 10.4686232833090289473352357524, 11.61032849190742468019644228619, 12.69734640587289330975806237971, 14.14198752947628214188633661294, 14.74529867823909196493859036251, 15.37324409427641461099537334843, 16.50806545349317347279147824896, 17.29219001348438661958621665497, 18.78869821611395987328455673321, 19.27207415158756096654362824663, 20.28675848072325951565866124612, 21.20856515660721592217866380033, 22.10375693899033083585923675244, 22.72130472552315867980366511812, 23.76157008116646807957752702199, 24.9511931082010971523429696925, 25.82197018825933482594367104285

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