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Properties

Label 2-2912-728.517-c0-0-0
Degree $2$
Conductor $2912$
Sign $0.716 - 0.697i$
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  + (−0.258 − 0.448i)3-s + 1.93i·5-s + (−0.866 − 0.5i)7-s + (0.366 − 0.633i)9-s + (0.965 − 0.258i)13-s + (0.866 − 0.499i)15-s + (1.22 + 0.707i)19-s + 0.517i·21-s + (0.5 + 0.866i)23-s − 2.73·25-s − 0.896·27-s + (0.965 − 1.67i)35-s + (−0.366 − 0.366i)39-s + (1.22 + 0.707i)45-s + (0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.258 − 0.448i)3-s + 1.93i·5-s + (−0.866 − 0.5i)7-s + (0.366 − 0.633i)9-s + (0.965 − 0.258i)13-s + (0.866 − 0.499i)15-s + (1.22 + 0.707i)19-s + 0.517i·21-s + (0.5 + 0.866i)23-s − 2.73·25-s − 0.896·27-s + (0.965 − 1.67i)35-s + (−0.366 − 0.366i)39-s + (1.22 + 0.707i)45-s + (0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.085731691\)
\(L(\frac12)\) \(\approx\) \(1.085731691\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.965 + 0.258i)T \)
good3 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - 1.93iT - T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.266601753816635361661018040567, −7.968257281949375083180385429361, −7.22951485719637951351957665761, −6.88316502313737933446269538795, −6.15333541277380750725350771878, −5.61103797937130246779345845281, −3.76655706470903250256124474777, −3.56947435867775701384421663499, −2.65498687989088194525879992723, −1.22643789177756366920485086909, 0.836656680435481799937838588787, 2.03108779678310087792979877616, 3.40853296499517909309958296635, 4.32583180302728783659650995304, 5.02040185714729634925816989592, 5.53499221861098202688016602914, 6.40267702883270662637690241042, 7.45190627752973967111896527286, 8.384822384901132038000683894438, 8.873774473634279365060724496203

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