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Properties

Label 2-2912-728.107-c0-0-0
Degree $2$
Conductor $2912$
Sign $0.325 + 0.945i$
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)3-s + (−0.866 − 0.5i)5-s + i·7-s + (0.5 − 0.866i)11-s + i·13-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)19-s + (0.866 + 0.5i)21-s − 2i·23-s + 27-s + (0.866 − 0.5i)29-s + (0.866 − 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.5 − 0.866i)35-s + (0.866 + 0.5i)39-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.866 − 0.5i)5-s + i·7-s + (0.5 − 0.866i)11-s + i·13-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)19-s + (0.866 + 0.5i)21-s − 2i·23-s + 27-s + (0.866 − 0.5i)29-s + (0.866 − 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.5 − 0.866i)35-s + (0.866 + 0.5i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :0),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286331313\)
\(L(\frac12)\) \(\approx\) \(1.286331313\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + 2iT - T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
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

−8.563710464218024460482049801764, −8.330504287396196183263762050154, −7.35671020097078313392102765696, −6.54399873664481130677910999428, −6.01255922708122516831342151166, −4.62416306669467965755958430644, −4.26507641774063281630102816012, −2.81509808379246081577271584049, −2.26162059233679385888283944106, −0.878020261799088288070335889632, 1.32146734246307972710866374836, 2.95507095246583913182854027884, 3.67846858697648895926639916634, 4.10407250123421276693561503817, 4.95221943254240720503036446232, 6.10427414464888157679500699422, 7.12238722221907455433834140589, 7.53808115022345223072637811516, 8.272368455364995129854571460781, 9.215925241002901953448161133114

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