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Properties

Label 2-2912-364.263-c0-0-1
Degree $2$
Conductor $2912$
Sign $0.998 + 0.0569i$
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  + (0.5 + 0.866i)5-s − i·7-s + 9-s − 13-s + (0.5 + 0.866i)17-s − 2i·19-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)35-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 + 0.866i)45-s + (−0.866 + 0.5i)47-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s − i·7-s + 9-s − 13-s + (0.5 + 0.866i)17-s − 2i·19-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)35-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 + 0.866i)45-s + (−0.866 + 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.998 + 0.0569i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ 0.998 + 0.0569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.456617111\)
\(L(\frac12)\) \(\approx\) \(1.456617111\)
\(L(1)\) 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 \( 1 + iT \)
13 \( 1 + T \)
good3 \( 1 - T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.289404069725070002885675245191, −7.83315755922405298025352735900, −7.42163479889185392946699608096, −6.73179803836719908628784791608, −6.13852496639072899830562639544, −4.83171873227696173427336332765, −4.37203430500965283752138901920, −3.21045877839861181455580413343, −2.41262670907915739341797048515, −1.12996429314413170921987416089, 1.28758956107251820057287433572, 2.18810254105662554123589555701, 3.27277079187133878002913402487, 4.46221523099855679242444146330, 5.15249817197963838890160080184, 5.69137873070247673921951691595, 6.67479588768798516581991149642, 7.52193007206578490058806444835, 8.244627852477459469401430698099, 9.042947459024283828757752366437

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