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Properties

Label 2-936-24.11-c1-0-25
Degree $2$
Conductor $936$
Sign $-0.866 + 0.499i$
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  + (−0.668 − 1.24i)2-s + (−1.10 + 1.66i)4-s − 3.68·5-s + 5.03i·7-s + (2.81 + 0.262i)8-s + (2.46 + 4.58i)10-s − 3.64i·11-s − i·13-s + (6.27 − 3.36i)14-s + (−1.55 − 3.68i)16-s + 5.67i·17-s − 1.84·19-s + (4.06 − 6.13i)20-s + (−4.54 + 2.43i)22-s + 2.33·23-s + ⋯
L(s)  = 1  + (−0.472 − 0.881i)2-s + (−0.552 + 0.833i)4-s − 1.64·5-s + 1.90i·7-s + (0.995 + 0.0926i)8-s + (0.778 + 1.45i)10-s − 1.09i·11-s − 0.277i·13-s + (1.67 − 0.900i)14-s + (−0.389 − 0.921i)16-s + 1.37i·17-s − 0.422·19-s + (0.909 − 1.37i)20-s + (−0.968 + 0.519i)22-s + 0.487·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.866 + 0.499i$
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: \((2,\ 936,\ (\ :1/2),\ -0.866 + 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0694384 - 0.259613i\)
\(L(\frac12)\) \(\approx\) \(0.0694384 - 0.259613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.668 + 1.24i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 3.68T + 5T^{2} \)
7 \( 1 - 5.03iT - 7T^{2} \)
11 \( 1 + 3.64iT - 11T^{2} \)
17 \( 1 - 5.67iT - 17T^{2} \)
19 \( 1 + 1.84T + 19T^{2} \)
23 \( 1 - 2.33T + 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 - 1.61iT - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + 1.54T + 43T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 - 2.41T + 53T^{2} \)
59 \( 1 + 7.69iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 1.62T + 67T^{2} \)
71 \( 1 + 6.50T + 71T^{2} \)
73 \( 1 - 7.47T + 73T^{2} \)
79 \( 1 - 5.87iT - 79T^{2} \)
83 \( 1 - 1.25iT - 83T^{2} \)
89 \( 1 + 1.11iT - 89T^{2} \)
97 \( 1 - 6.82T + 97T^{2} \)
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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

−9.610875138325462789574577815121, −8.631385364985330014293694742251, −8.404814608242793658344250489634, −7.65277036086278004787331578921, −6.21431149677773620157670724227, −5.18213532924033872914415737073, −3.91661636065874907234802705488, −3.26036863181969421017946509054, −2.10805921093745853397663319972, −0.17829647723652570688089415259, 1.11843832293778828533343480284, 3.47394413018979605437110874158, 4.49048113388987995215203443556, 4.80303484515119854561605734256, 6.70916049768110744480116439259, 7.24552941036060051173665577478, 7.55522498825104949186062550731, 8.493388579735291671403152966788, 9.522903055749232757237427138280, 10.33788929331243897477515426560

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