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Properties

Label 1-2960-2960.1483-r0-0-0
Degree $1$
Conductor $2960$
Sign $0.994 - 0.103i$
Analytic cond. $13.7461$
Root an. cond. $13.7461$
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  + (−0.173 + 0.984i)3-s + (0.642 − 0.766i)7-s + (−0.939 − 0.342i)9-s + (−0.866 + 0.5i)11-s + (0.939 − 0.342i)13-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)19-s + (0.642 + 0.766i)21-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)27-s + (−0.866 + 0.5i)29-s + 31-s + (−0.342 − 0.939i)33-s + (0.173 + 0.984i)39-s + (0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)3-s + (0.642 − 0.766i)7-s + (−0.939 − 0.342i)9-s + (−0.866 + 0.5i)11-s + (0.939 − 0.342i)13-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)19-s + (0.642 + 0.766i)21-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)27-s + (−0.866 + 0.5i)29-s + 31-s + (−0.342 − 0.939i)33-s + (0.173 + 0.984i)39-s + (0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.994 - 0.103i$
Analytic conductor: \(13.7461\)
Root analytic conductor: \(13.7461\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (0:\ ),\ 0.994 - 0.103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.622291793 - 0.08437385805i\)
\(L(\frac12)\) \(\approx\) \(1.622291793 - 0.08437385805i\)
\(L(1)\) \(\approx\) \(1.078390537 + 0.1474485419i\)
\(L(1)\) \(\approx\) \(1.078390537 + 0.1474485419i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (0.642 - 0.766i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (-0.642 - 0.766i)T \)
61 \( 1 + (-0.342 - 0.939i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−18.865303384250478472570513770120, −18.48454059659961196283153762223, −17.89448871435946453979641078972, −17.14705170952798289735298125349, −16.372340538519447516955634566964, −15.60409527609445817713608753236, −14.82789039470993751956835809891, −14.0943307048904555904290821239, −13.35353538980653074741495966456, −12.83796406937265061367384500683, −12.03817937453125493452596394240, −11.24184764537767545695808943293, −10.98108090058071734497617660482, −9.7901716061583320994090487771, −8.766530964229581221374555151660, −8.24549564832505764926068370592, −7.72131202451870875208644631649, −6.72298112536699995440620389944, −5.92733220531455040921557794818, −5.47299662906269736208484175116, −4.56513217312850643416751052129, −3.28203888657370055774040898786, −2.62259042808796570148185776857, −1.671639886858006565983047937293, −0.96801616876378307990736286471, 0.61480547410317984422274404113, 1.6596587973739573427828083654, 3.02107603796592307239642723789, 3.45596951523695204109593903162, 4.53109163719011101382044767874, 5.07109077220552925295633137296, 5.669589853394730239657801985, 6.803808721903394749451478666550, 7.672263704823852550267570004125, 8.21848250469098742296814827847, 9.274881588838904391739386239890, 9.81809025918516052742819422110, 10.618917357837003728822936556605, 11.12927362187508424659799750766, 11.736632062564019941310526998584, 12.77011610586548299669420813064, 13.67666701276848653162299148010, 14.10295918997300815898710303534, 15.07845311667574579575408530729, 15.55538949608436798782796154905, 16.2992685302409223348322579308, 16.84250513629663654528345716829, 17.79733499358268820865271828387, 18.079393275753710238328486845041, 19.06520384955677163526027917973

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