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Properties

Label 2-2548-364.111-c0-0-1
Degree $2$
Conductor $2548$
Sign $0.978 + 0.208i$
Analytic cond. $1.27161$
Root an. cond. $1.12766$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.184 + 0.184i)5-s + (0.707 − 0.707i)8-s + (0.5 + 0.866i)9-s + (−0.130 + 0.226i)10-s + (0.608 + 0.793i)13-s + (0.500 − 0.866i)16-s + (−0.608 − 1.05i)17-s + (0.707 + 0.707i)18-s + (−0.0675 + 0.252i)20-s + 0.931i·25-s + (0.793 + 0.608i)26-s + (0.258 − 0.448i)29-s + (0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.184 + 0.184i)5-s + (0.707 − 0.707i)8-s + (0.5 + 0.866i)9-s + (−0.130 + 0.226i)10-s + (0.608 + 0.793i)13-s + (0.500 − 0.866i)16-s + (−0.608 − 1.05i)17-s + (0.707 + 0.707i)18-s + (−0.0675 + 0.252i)20-s + 0.931i·25-s + (0.793 + 0.608i)26-s + (0.258 − 0.448i)29-s + (0.258 − 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.978 + 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.292674283\)
\(L(\frac12)\) \(\approx\) \(2.292674283\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 \)
13 \( 1 + (-0.608 - 0.793i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.184 - 0.184i)T - iT^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.608 + 1.05i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.410 - 1.53i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 1.73T + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.245250991664252544174610044448, −8.100833063730528848753429497776, −7.30690316424064381822575072155, −6.73481560054053262983228836555, −5.87822731695494956529503793163, −4.90843736705058326320809544194, −4.39484117882939288653672773026, −3.43964950449349341792125575124, −2.45424121669955621200324240691, −1.51844113104382788953936337615, 1.38887047715990225087939611703, 2.69111800199753915294669391213, 3.70173956150391196687115703889, 4.20322916162922340213099339351, 5.19135716560475933365734753647, 6.08129625879497331391077860247, 6.59069627022979877797964223223, 7.42311075144956169453659138274, 8.334853675728258589068400387473, 8.799331404933719405821564663043

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