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Properties

Label 2-2548-364.111-c0-0-2
Degree $2$
Conductor $2548$
Sign $0.865 + 0.500i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.865 + 0.500i$
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: \((2,\ 2548,\ (\ :0),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.383182055\)
\(L(\frac12)\) \(\approx\) \(2.383182055\)
\(L(1)\) 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

−9.150825525135411366664492608745, −7.934407616769289126257564156063, −7.52796882473461399750517370001, −6.58039155245658762040791347939, −5.60291131179087117677358662706, −5.19232525702027028052424034735, −4.25044107720434093429443036755, −3.41000499044606817430082900438, −2.36076173224022889579879225146, −1.45718536794495802937812014644, 1.55451908078822103091034375195, 2.77362387113456945457298714505, 3.48092762179773153347665174630, 4.59098083132214290391928689607, 4.99664895561630804619079288106, 6.31716494482403979784694543166, 6.55122652779422036147745038168, 7.43206209657606342351134292775, 8.131870632927564623407105542997, 9.281450512882368641167055263354

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