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Properties

Label 2-936-117.94-c1-0-36
Degree $2$
Conductor $936$
Sign $0.359 + 0.932i$
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  + (1.54 − 0.778i)3-s + (0.871 − 1.50i)5-s + 1.46·7-s + (1.78 − 2.40i)9-s + (0.809 − 1.40i)11-s + (−3.60 + 0.143i)13-s + (0.172 − 3.01i)15-s + (0.868 − 1.50i)17-s + (1.23 − 2.13i)19-s + (2.26 − 1.14i)21-s − 0.926·23-s + (0.981 + 1.70i)25-s + (0.887 − 5.11i)27-s + (−1.15 + 2.00i)29-s + (−1.56 + 2.71i)31-s + ⋯
L(s)  = 1  + (0.893 − 0.449i)3-s + (0.389 − 0.674i)5-s + 0.554·7-s + (0.595 − 0.803i)9-s + (0.243 − 0.422i)11-s + (−0.999 + 0.0398i)13-s + (0.0445 − 0.778i)15-s + (0.210 − 0.364i)17-s + (0.282 − 0.489i)19-s + (0.495 − 0.249i)21-s − 0.193·23-s + (0.196 + 0.340i)25-s + (0.170 − 0.985i)27-s + (−0.215 + 0.372i)29-s + (−0.281 + 0.487i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.359 + 0.932i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.359 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96871 - 1.35057i\)
\(L(\frac12)\) \(\approx\) \(1.96871 - 1.35057i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 + 0.778i)T \)
13 \( 1 + (3.60 - 0.143i)T \)
good5 \( 1 + (-0.871 + 1.50i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
11 \( 1 + (-0.809 + 1.40i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.868 + 1.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.23 + 2.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.926T + 23T^{2} \)
29 \( 1 + (1.15 - 2.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.56 - 2.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.19 - 2.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.49T + 41T^{2} \)
43 \( 1 + 4.52T + 43T^{2} \)
47 \( 1 + (-0.848 - 1.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + (1.84 + 3.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 0.988T + 61T^{2} \)
67 \( 1 + 7.48T + 67T^{2} \)
71 \( 1 + (-2.65 + 4.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.51T + 73T^{2} \)
79 \( 1 + (1.27 + 2.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.73 - 2.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.05 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−9.533764721397447463744573340904, −9.126113193558662380405178332109, −8.250999347587482665010154440139, −7.49694136782383754374575118337, −6.67346208193867061537895485909, −5.41932307485112622389963051602, −4.62195118050876530889177305248, −3.36289485361598546476167574498, −2.24557548127243180443805679324, −1.10408206727923148807813530198, 1.86667737788122144810023039836, 2.72061344499908092492543069954, 3.89043916539777926593367599690, 4.78400059425706527463371256786, 5.85376443680765857759472457381, 7.07417375166615147929921952761, 7.70614551593135430409985100981, 8.554706093234970332614570156774, 9.557577743377967870929901283558, 10.06742788084573297819877709702

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