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Properties

Label 1-2960-2960.1203-r1-0-0
Degree $1$
Conductor $2960$
Sign $0.765 - 0.643i$
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  + (−0.173 + 0.984i)3-s + (0.642 − 0.766i)7-s + (−0.939 − 0.342i)9-s + (−0.866 + 0.5i)11-s + (0.342 + 0.939i)13-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)19-s + (0.642 + 0.766i)21-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.342 − 0.939i)33-s + (−0.984 + 0.173i)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 + (0.342 + 0.939i)13-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)19-s + (0.642 + 0.766i)21-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.342 − 0.939i)33-s + (−0.984 + 0.173i)39-s + (−0.939 + 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.765 - 0.643i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ 0.765 - 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8333040577 - 0.3036453621i\)
\(L(\frac12)\) \(\approx\) \(0.8333040577 - 0.3036453621i\)
\(L(1)\) \(\approx\) \(0.8425460072 + 0.2494383124i\)
\(L(1)\) \(\approx\) \(0.8425460072 + 0.2494383124i\)

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 \( 1 + (0.342 + 0.939i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.642 + 0.766i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.86861538732733898144384906879, −18.282493840435875568960692836489, −17.626170859254820825488197731675, −17.3025287320945007434501057730, −16.14891151611467648736878182607, −15.28859032638829649914242486428, −15.03321033391147117621114690016, −13.74883110808186494054814587018, −13.34425389774228245550702152888, −12.78025007442071169957349772594, −11.898142448094990969416798790451, −11.15206935224573665367566767941, −10.86190689349498716988411942017, −9.6181360631786063302157851535, −8.626521306772234633171380039614, −8.218523862685951174692712577630, −7.554819873315680286095301672831, −6.621197126080662774362928686679, −5.82009472959369875284204319026, −5.337728623322771030292183924768, −4.44270860279991013467096959972, −3.030027296981231352888916431470, −2.55042309040372200303494225527, −1.66088271103864232214620402, −0.67358749877892134650670107577, 0.19154063032959742372623381661, 1.45158945389622182062412352683, 2.3933704517650959071041812979, 3.43803509627574531863438697267, 4.21845603530888371849863724984, 4.82025169970914292308081467562, 5.38737280613604151580099731572, 6.62660479097954256628557748300, 7.08749794603535043277767483858, 8.40556697637718767515754953869, 8.598458555183130927410524053808, 9.752458794849905281684309117185, 10.35590108487824533146277875707, 10.884386857622097925737659845521, 11.56027750106717949970703416988, 12.376916159637291447999247913678, 13.331799385706411693609311681953, 14.10811609891845588833138296155, 14.61737841030374540364510534506, 15.3810236075695681469879216607, 16.15732696071961668780776845327, 16.61312455817288982580185958713, 17.36078013346275585447188505256, 18.07399224578209476725169212480, 18.67836688086655420293634634967

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