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Properties

Label 2-2960-740.379-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.977 - 0.208i$
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  + (0.5 − 0.866i)5-s + (0.939 + 0.342i)9-s + (0.592 + 1.62i)13-s + (−0.439 + 1.20i)17-s + (−0.499 − 0.866i)25-s + (0.939 + 1.62i)29-s + (−0.939 + 0.342i)37-s + (1.43 − 0.524i)41-s + (0.766 − 0.642i)45-s + (−0.173 − 0.984i)49-s + (−1.11 − 1.32i)53-s + (−1.76 + 0.642i)61-s + (1.70 + 0.300i)65-s − 1.73i·73-s + (0.766 + 0.642i)81-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (0.939 + 0.342i)9-s + (0.592 + 1.62i)13-s + (−0.439 + 1.20i)17-s + (−0.499 − 0.866i)25-s + (0.939 + 1.62i)29-s + (−0.939 + 0.342i)37-s + (1.43 − 0.524i)41-s + (0.766 − 0.642i)45-s + (−0.173 − 0.984i)49-s + (−1.11 − 1.32i)53-s + (−1.76 + 0.642i)61-s + (1.70 + 0.300i)65-s − 1.73i·73-s + (0.766 + 0.642i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.977 - 0.208i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.977 - 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.481864863\)
\(L(\frac12)\) \(\approx\) \(1.481864863\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.939 - 0.342i)T \)
good3 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \)
19 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.848783161453405483665834183237, −8.480794721828214663694714985375, −7.39705352333432086953325069025, −6.60785856572040316126483361924, −6.02169852799747054102831063600, −4.86490024805418520897403517120, −4.44381553768921732595813673864, −3.52770570051339012888827305077, −1.92479720447447357976641160795, −1.49358667221553987625071396665, 1.06162985192457472635288034548, 2.47251635465140625155527843455, 3.13365276779110987950459235299, 4.14452673835279198071711663017, 5.08267186395811083334213609417, 6.06396291775232058139535177913, 6.49753384360638582715103577026, 7.53138318362009678337159091987, 7.86283374976500743090629282043, 9.113088016783039748765909611855

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