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Properties

Label 2-2960-185.154-c0-0-1
Degree $2$
Conductor $2960$
Sign $0.763 - 0.646i$
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  + 3-s + 5-s + i·7-s + i·11-s + 15-s + i·21-s + 25-s − 27-s + (−1 + i)31-s + i·33-s + i·35-s − i·37-s − i·41-s + (1 − i)43-s − i·47-s + ⋯
L(s)  = 1  + 3-s + 5-s + i·7-s + i·11-s + 15-s + i·21-s + 25-s − 27-s + (−1 + i)31-s + i·33-s + i·35-s − i·37-s − i·41-s + (1 − i)43-s − i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.763 - 0.646i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.763 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.972410737\)
\(L(\frac12)\) \(\approx\) \(1.972410737\)
\(L(1)\) 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 \( 1 - T \)
37 \( 1 + iT \)
good3 \( 1 - T + T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (1 + i)T + iT^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.059475940381516885011002168660, −8.554746379815246561482071674591, −7.50542736051155391613598042313, −6.89223291455868412644307205074, −5.72944983149797268441627081549, −5.41707648081504846016104737054, −4.24333617522462865527958488157, −3.16811921201409471414787485269, −2.31855443981185852353337202802, −1.83618408207245179456766491988, 1.16579340363449664436353293054, 2.33499886408065518786963911309, 3.14461770674360719510883973381, 3.90586087607781531562269818988, 4.95897654898646017536991845606, 5.92413727805773491659101914419, 6.50641991675492152299541243899, 7.55514317719899712486690314678, 8.103330691581431595383593909047, 8.885091469160652560019143267477

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