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Properties

Label 2-2960-740.483-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.873 - 0.486i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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)5-s + (0.984 − 0.173i)9-s + (−0.173 + 0.984i)13-s + (0.673 − 0.118i)17-s + (0.499 + 0.866i)25-s + (−1.10 − 0.296i)29-s + (−0.984 − 0.173i)37-s + (0.673 + 0.118i)41-s + (0.939 + 0.342i)45-s + (−0.642 − 0.766i)49-s + (0.218 + 0.469i)53-s + (1.34 − 0.939i)61-s + (−0.642 + 0.766i)65-s + (−1.36 + 1.36i)73-s + (0.939 − 0.342i)81-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)5-s + (0.984 − 0.173i)9-s + (−0.173 + 0.984i)13-s + (0.673 − 0.118i)17-s + (0.499 + 0.866i)25-s + (−1.10 − 0.296i)29-s + (−0.984 − 0.173i)37-s + (0.673 + 0.118i)41-s + (0.939 + 0.342i)45-s + (−0.642 − 0.766i)49-s + (0.218 + 0.469i)53-s + (1.34 − 0.939i)61-s + (−0.642 + 0.766i)65-s + (−1.36 + 1.36i)73-s + (0.939 − 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.873 - 0.486i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.873 - 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.573306565\)
\(L(\frac12)\) \(\approx\) \(1.573306565\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.984 + 0.173i)T \)
good3 \( 1 + (-0.984 + 0.173i)T^{2} \)
7 \( 1 + (0.642 + 0.766i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.673 + 0.118i)T + (0.939 - 0.342i)T^{2} \)
19 \( 1 + (0.984 - 0.173i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.10 + 0.296i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.218 - 0.469i)T + (-0.642 + 0.766i)T^{2} \)
59 \( 1 + (0.642 - 0.766i)T^{2} \)
61 \( 1 + (-1.34 + 0.939i)T + (0.342 - 0.939i)T^{2} \)
67 \( 1 + (-0.642 - 0.766i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 + (-0.642 - 0.766i)T^{2} \)
83 \( 1 + (0.342 + 0.939i)T^{2} \)
89 \( 1 + (-0.766 + 0.357i)T + (0.642 - 0.766i)T^{2} \)
97 \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.275790455499943533205560930411, −8.231535434962728837237741468494, −7.19202639018795899075174604876, −6.89733322856523418033557720434, −5.96586223248826515889001607974, −5.22794028202670951552704914141, −4.24847031935696059343149325791, −3.41811914829072171782829983126, −2.25242539207637462439014559111, −1.46520827496358403023031531974, 1.15159329273388064388630861395, 2.08855582909540551692599110117, 3.22113463740069736553701185660, 4.22976981834665612215454330887, 5.16695784149325676186416525466, 5.64895171309824815584429818298, 6.57881064224298950246023687956, 7.43677080602468869278933064159, 8.061049900654286218003121813536, 8.987161520235161241658196151846

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