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Properties

Label 2-2912-364.51-c0-0-4
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 + 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 + (0.5 + 0.866i)31-s + (−0.866 − 0.499i)33-s + (−0.866 − 0.5i)35-s + (0.866 + 0.5i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s − 7-s + (−0.5 − 0.866i)11-s + 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 + (0.5 + 0.866i)31-s + (−0.866 − 0.499i)33-s + (−0.866 − 0.5i)35-s + (0.866 + 0.5i)37-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: $\chi_{2912} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ 0.980 + 0.197i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.793662483\)
\(L(\frac12)\) \(\approx\) \(1.793662483\)
\(L(1)\) 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 \( 1 - T \)
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 \( 1 + T^{2} \)
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 \( 1 + 2iT - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
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

−8.774826768965038875730379616188, −8.380430521001953947515351909706, −7.31868673237813645620368146820, −6.75492250276480059339971713402, −5.86839170944046404735763822395, −5.41174401704239908415773533343, −3.80766951069659873015248468099, −3.05133946010759103296165880554, −2.53912568256000408551789120393, −1.30031785491693215899853259240, 1.29441096137028170791807603606, 2.64444053393142533621233520977, 3.18060870956564645310967425638, 4.17894766210980900163847810019, 5.05499444763161877870077695721, 5.99211816615426140390075657843, 6.52457744874554858365258660960, 7.69490630327938864034302316571, 8.279792786408509436428531081374, 9.277904040586998899938514996134

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