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Properties

Label 1-2960-2960.29-r1-0-0
Degree $1$
Conductor $2960$
Sign $0.0986 + 0.995i$
Analytic cond. $318.096$
Root an. cond. $318.096$
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  + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s − i·11-s + (0.5 + 0.866i)13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.5i)21-s − i·23-s − i·27-s − 29-s + i·31-s + (0.5 + 0.866i)33-s + (0.866 + 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s − i·11-s + (0.5 + 0.866i)13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.5i)21-s − i·23-s − i·27-s − 29-s + i·31-s + (0.5 + 0.866i)33-s + (0.866 + 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0986 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0986 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.0986 + 0.995i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ 0.0986 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6397038881 + 0.5794431681i\)
\(L(\frac12)\) \(\approx\) \(0.6397038881 + 0.5794431681i\)
\(L(1)\) \(\approx\) \(1.126551283 - 0.2587896707i\)
\(L(1)\) \(\approx\) \(1.126551283 - 0.2587896707i\)

Euler product

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

−18.94903507138997348520777951223, −18.272009153826314644530044334975, −17.310285708492027228090128939404, −16.41296414300189328150307018355, −15.878130458032372747654137131695, −15.19814932858405642140179015429, −14.77103260364025796678974093645, −13.77795545051592139421277043604, −13.11804816877102357833181732430, −12.71885725875699280168028378606, −11.459115910771049299578651429606, −10.94872171085565548241886584210, −10.02720741689872381061064926106, −9.41870457415848596475596955593, −8.60691840462200579384245338412, −8.23538278856053809044484563151, −7.38032867357650238803755954022, −6.05703621651263593674511732849, −5.82389160219270190563267274409, −4.710790118585767989683559075811, −3.6597588350237876236370370846, −3.29782821155389343737578093062, −2.36104693463487553590073292075, −1.55694202517964273282912089478, −0.12372340508887812438488135102, 0.88478970046787440185927495425, 1.91440658302027011781604472729, 2.50719395497700162501390846076, 3.58617677931830846334015408324, 4.20463888997950139834251761924, 4.93311438860592112215004977083, 6.49063667221807353059815623142, 6.7865442727075247639741612341, 7.367528335607839057939085236297, 8.348948337433807773010510111743, 9.090670844539568059936368807021, 9.58815505335341928622729201096, 10.49384409931758249552194299112, 11.2457934781828286144088164522, 12.20431401463279880020783603463, 13.00119947671936230054906679472, 13.3127467099887681010198303471, 14.22624734133718299693552279271, 14.635801508767201741168497130034, 15.65638560754416682799478114462, 16.09228920027345860924001330417, 17.1622323141135555247525058815, 17.70197510488210466813506170861, 18.49154248899668117570908668271, 19.190621184230576929225873564970

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