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Properties

Label 2-936-24.11-c1-0-42
Degree $2$
Conductor $936$
Sign $0.110 + 0.993i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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  + (1.32 − 0.492i)2-s + (1.51 − 1.30i)4-s + 0.678·5-s − 4.56i·7-s + (1.36 − 2.47i)8-s + (0.899 − 0.333i)10-s + 6.10i·11-s − i·13-s + (−2.24 − 6.04i)14-s + (0.593 − 3.95i)16-s − 6.72i·17-s − 1.27·19-s + (1.02 − 0.884i)20-s + (3.00 + 8.09i)22-s + 5.10·23-s + ⋯
L(s)  = 1  + (0.937 − 0.348i)2-s + (0.757 − 0.652i)4-s + 0.303·5-s − 1.72i·7-s + (0.483 − 0.875i)8-s + (0.284 − 0.105i)10-s + 1.84i·11-s − 0.277i·13-s + (−0.599 − 1.61i)14-s + (0.148 − 0.988i)16-s − 1.63i·17-s − 0.292·19-s + (0.229 − 0.197i)20-s + (0.640 + 1.72i)22-s + 1.06·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18761 - 1.95732i\)
\(L(\frac12)\) \(\approx\) \(2.18761 - 1.95732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.492i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 - 0.678T + 5T^{2} \)
7 \( 1 + 4.56iT - 7T^{2} \)
11 \( 1 - 6.10iT - 11T^{2} \)
17 \( 1 + 6.72iT - 17T^{2} \)
19 \( 1 + 1.27T + 19T^{2} \)
23 \( 1 - 5.10T + 23T^{2} \)
29 \( 1 - 1.72T + 29T^{2} \)
31 \( 1 - 2.37iT - 31T^{2} \)
37 \( 1 - 6.99iT - 37T^{2} \)
41 \( 1 - 2.28iT - 41T^{2} \)
43 \( 1 - 2.68T + 43T^{2} \)
47 \( 1 - 5.32T + 47T^{2} \)
53 \( 1 + 6.35T + 53T^{2} \)
59 \( 1 + 4.46iT - 59T^{2} \)
61 \( 1 + 6.35iT - 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 5.82T + 73T^{2} \)
79 \( 1 - 12.6iT - 79T^{2} \)
83 \( 1 - 9.84iT - 83T^{2} \)
89 \( 1 - 15.8iT - 89T^{2} \)
97 \( 1 - 6.95T + 97T^{2} \)
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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

−9.857081642418350389970201013695, −9.635962726265894727885228979350, −7.78060428262938178039588183412, −7.07833416157161449472865608645, −6.61841282595976718130022909853, −5.05589409446255914955709144944, −4.62435535672288736337036689622, −3.62178426169677356390177715043, −2.40198121016383759669159948417, −1.08450339992817448367512475856, 1.98641252123063422129083094984, 2.97196423693872150133268103446, 3.94736977748370873528302816479, 5.30900734121540983065796235795, 5.95267185802674554249701990280, 6.29417668131908940838102394884, 7.76039409328199735831197956974, 8.645422951800763457726032422236, 9.006624087486906354226348350423, 10.53798298527122727989452576214

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