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Properties

Label 2-2912-728.237-c0-0-1
Degree $2$
Conductor $2912$
Sign $-0.265 - 0.964i$
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 + 1.5i)3-s + 1.73·5-s + (−0.5 + 0.866i)7-s + (−1 + 1.73i)9-s + (−0.866 − 0.5i)13-s + (1.49 + 2.59i)15-s − 1.73·21-s + (−0.5 − 0.866i)23-s + 1.99·25-s − 1.73·27-s + (−0.866 + 1.49i)35-s − 1.73i·39-s + (−1.73 + 3i)45-s + (−0.499 − 0.866i)49-s + (0.866 − 1.5i)59-s + ⋯
L(s)  = 1  + (0.866 + 1.5i)3-s + 1.73·5-s + (−0.5 + 0.866i)7-s + (−1 + 1.73i)9-s + (−0.866 − 0.5i)13-s + (1.49 + 2.59i)15-s − 1.73·21-s + (−0.5 − 0.866i)23-s + 1.99·25-s − 1.73·27-s + (−0.866 + 1.49i)35-s − 1.73i·39-s + (−1.73 + 3i)45-s + (−0.499 − 0.866i)49-s + (0.866 − 1.5i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ -0.265 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.995918755\)
\(L(\frac12)\) \(\approx\) \(1.995918755\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - 1.73T + T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + T^{2} \)
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

−9.378737243842838134821500394295, −8.739038965987741477885138331089, −8.014830901508172259172447647293, −6.70425312138332864163044755591, −5.87777850574482344496670255011, −5.25390642127094472153210000225, −4.62025652308720266794180091224, −3.41566279771880274089542340131, −2.61160197834719729589001989674, −2.12201618605801013601391304357, 1.17535486519201918524852363376, 2.01550396325368811208157216910, 2.65392681386733528411163941868, 3.66994068718345509775515915336, 5.01717781447296214306684575359, 6.02071818518047786439042659393, 6.54856986317789257674365671671, 7.19289330498016057017110695247, 7.78089523790443356982464722432, 8.768387412106809168137594974680

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