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Properties

Label 2-2496-24.11-c1-0-70
Degree $2$
Conductor $2496$
Sign $0.582 + 0.813i$
Analytic cond. $19.9306$
Root an. cond. $4.46437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.62 − 0.609i)3-s + 1.44·5-s − 3.46i·7-s + (2.25 − 1.97i)9-s + 6.00i·11-s + i·13-s + (2.33 − 0.878i)15-s − 4.92i·17-s + 1.73·19-s + (−2.11 − 5.62i)21-s + 4.37·23-s − 2.92·25-s + (2.45 − 4.57i)27-s − 0.607·29-s − 8.83i·31-s + ⋯
L(s)  = 1  + (0.936 − 0.351i)3-s + 0.644·5-s − 1.31i·7-s + (0.752 − 0.658i)9-s + 1.81i·11-s + 0.277i·13-s + (0.603 − 0.226i)15-s − 1.19i·17-s + 0.398·19-s + (−0.461 − 1.22i)21-s + 0.912·23-s − 0.584·25-s + (0.472 − 0.881i)27-s − 0.112·29-s − 1.58i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.582 + 0.813i$
Analytic conductor: \(19.9306\)
Root analytic conductor: \(4.46437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2496,\ (\ :1/2),\ 0.582 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.015090426\)
\(L(\frac12)\) \(\approx\) \(3.015090426\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 0.609i)T \)
13 \( 1 - iT \)
good5 \( 1 - 1.44T + 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 - 6.00iT - 11T^{2} \)
17 \( 1 + 4.92iT - 17T^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 - 4.37T + 23T^{2} \)
29 \( 1 + 0.607T + 29T^{2} \)
31 \( 1 + 8.83iT - 31T^{2} \)
37 \( 1 + 3.88iT - 37T^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 - 8.29T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 + 6.64iT - 61T^{2} \)
67 \( 1 + 5.83T + 67T^{2} \)
71 \( 1 + 5.38T + 71T^{2} \)
73 \( 1 + 4.76T + 73T^{2} \)
79 \( 1 - 0.230iT - 79T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 + 0.471iT - 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−9.052268501829083001538159343878, −7.67016888258599781509548761069, −7.38215254911109009219734140616, −6.89202474481715149417660205155, −5.75660560466001275654341106932, −4.48488461276300818979381324649, −4.14531710953164249455856298619, −2.82824857548935656601393720316, −2.03449163026419945395127612289, −0.977394140745723089103988651998, 1.38396934291891671936862822682, 2.52447683449900597860138185352, 3.13476350813081857792263888398, 4.02836664033089157463748392366, 5.49091451562725602586191106076, 5.60115372536887376182560624167, 6.65592416900329732037941162708, 7.78511208757365463226521626599, 8.638365075837695098323543142284, 8.808376329224134696800031763780

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