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Properties

Label 2-2912-364.207-c0-0-6
Degree $2$
Conductor $2912$
Sign $0.980 - 0.197i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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)3-s + (0.866 − 0.5i)5-s + 7-s + (0.5 − 0.866i)11-s + i·13-s + 0.999·15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.866 + 0.5i)21-s + (−0.866 + 0.5i)23-s − i·27-s − 2·29-s + (0.5 − 0.866i)31-s + (0.866 − 0.499i)33-s + (0.866 − 0.5i)35-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 − 0.5i)5-s + 7-s + (0.5 − 0.866i)11-s + i·13-s + 0.999·15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.866 + 0.5i)21-s + (−0.866 + 0.5i)23-s − i·27-s − 2·29-s + (0.5 − 0.866i)31-s + (0.866 − 0.499i)33-s + (0.866 − 0.5i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.980 - 0.197i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (1663, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ 0.980 - 0.197i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.094946112\)
\(L(\frac12)\) \(\approx\) \(2.094946112\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.969134575053662457061372958897, −8.372024633419994478661077583670, −7.80742330946187367100956210111, −6.57487708145658140561662682202, −5.81749557914666447740668170890, −5.15255679217789018183585595725, −3.93212762333108269244091816760, −3.68760354799865090219227041021, −2.10623035627316213819914395358, −1.60820682268310965815254353686, 1.52734740725178002614643111152, 2.29873387399325885997838322545, 2.91907156179074503086750185718, 4.19603858426406850125812354062, 5.14144727993664134761119421265, 5.80292943167052423246847037577, 6.99640014405156354419575044044, 7.35045671427616433009115480403, 8.135170178882450077189185541842, 8.905315183355627982388546669270

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