Properties

Label 2-2960-2960.2357-c0-0-0
Degree $2$
Conductor $2960$
Sign $-0.0137 - 0.999i$
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)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999·6-s + (0.366 + 1.36i)7-s + 0.999i·8-s − 0.999·10-s + (1 + i)11-s + (−0.866 + 0.499i)12-s + (0.866 + 0.5i)13-s + (−1 − 0.999i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.866 − 0.499i)20-s + (0.366 − 1.36i)21-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999·6-s + (0.366 + 1.36i)7-s + 0.999i·8-s − 0.999·10-s + (1 + i)11-s + (−0.866 + 0.499i)12-s + (0.866 + 0.5i)13-s + (−1 − 0.999i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.866 − 0.499i)20-s + (0.366 − 1.36i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.0137 - 0.999i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ -0.0137 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7849961108\)
\(L(\frac12)\) \(\approx\) \(0.7849961108\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-1 - i)T + iT^{2} \)
13 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + T + T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.088768288201378821917971293003, −8.623736167333531574001745865788, −7.32214284153172217799725399058, −6.78126187168490535282419685989, −6.25899944980471422820934320094, −5.52920115232650616021372752894, −5.06760941968196732492695794686, −3.34731282323365475947121039591, −1.88560717501294753247164714167, −1.59342331609734492688552362480, 0.802777985327476329299920031382, 1.53891220872824419082944388300, 3.07896274547623342882684593816, 3.98535081963709606700404993358, 4.78170971433015563940957216927, 5.76883465240412059866170189786, 6.49243069843905990273775438129, 7.20224763848325864685107427029, 8.452999639850065202727594773213, 8.618025208900398850906023694596

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