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Properties

Label 2-2960-185.179-c0-0-0
Degree $2$
Conductor $2960$
Sign $-0.646 + 0.763i$
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 − i·5-s + i·7-s − i·11-s + i·15-s − i·21-s − 25-s + 27-s + (−1 − i)31-s + i·33-s + 35-s − i·37-s + i·41-s + (−1 − i)43-s − i·47-s + ⋯
L(s)  = 1  − 3-s − i·5-s + i·7-s − i·11-s + i·15-s − i·21-s − 25-s + 27-s + (−1 − i)31-s + i·33-s + 35-s − i·37-s + i·41-s + (−1 − i)43-s − i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.646 + 0.763i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ -0.646 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4659341180\)
\(L(\frac12)\) \(\approx\) \(0.4659341180\)
\(L(1)\) 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 \( 1 + iT \)
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

−8.678138912802137436126231872316, −8.128680282000543430201327862612, −7.04092414068731602341612599756, −5.87397406091283897161096183959, −5.77658773910378469798921689824, −5.04313855943985713538836514320, −4.08938326542885783086393827808, −2.95433699839723432261760017877, −1.73761473197157885847033641072, −0.33469002145740885266890933020, 1.45177036608733505871314145623, 2.75346435835196292032892571803, 3.72426876535623464523948216503, 4.60889904539865313522161807224, 5.39139363290322717121176680772, 6.31884979598993498018886589014, 6.90815646013958289820178704103, 7.39054367692512863470855136988, 8.294977475313306523069875504684, 9.430521449211641066472587314416

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