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Properties

Label 2-2496-312.11-c0-0-1
Degree $2$
Conductor $2496$
Sign $0.733 + 0.679i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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 + (−1.86 + 0.5i)7-s + (−0.499 + 0.866i)9-s + (0.866 + 0.5i)13-s + (1.36 − 0.366i)19-s + (1.36 + 1.36i)21-s − i·25-s + 0.999·27-s + (0.366 − 0.366i)31-s + (0.366 − 1.36i)37-s − 0.999i·39-s + (0.866 + 0.5i)43-s + (2.36 − 1.36i)49-s + (−1 − 0.999i)57-s + (0.866 + 0.5i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−1.86 + 0.5i)7-s + (−0.499 + 0.866i)9-s + (0.866 + 0.5i)13-s + (1.36 − 0.366i)19-s + (1.36 + 1.36i)21-s − i·25-s + 0.999·27-s + (0.366 − 0.366i)31-s + (0.366 − 1.36i)37-s − 0.999i·39-s + (0.866 + 0.5i)43-s + (2.36 − 1.36i)49-s + (−1 − 0.999i)57-s + (0.866 + 0.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.733 + 0.679i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :0),\ 0.733 + 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8013775784\)
\(L(\frac12)\) \(\approx\) \(0.8013775784\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + iT^{2} \)
7 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
37 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.094480622767696593072879644616, −8.215860757974731141746168197296, −7.25960018554235475523643701401, −6.64873087947465132330104240055, −6.02179963708753159197057538521, −5.47567483502994902869027638851, −4.13553890013918983109963986241, −3.10740644012042756982932313421, −2.32578131289698046109945833271, −0.796091737994580609564721380774, 0.927182689337368569622974117805, 3.12413015353144234682857047330, 3.36491607317856018552632056527, 4.30430535345994512769822997956, 5.44874061710543987517729284758, 6.01436652217665322518220638409, 6.73694486964280175117597987817, 7.53813988414025027882466997875, 8.715114885569306665914973715594, 9.361820811914701854981865070927

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