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Properties

Label 2-936-104.101-c1-0-12
Degree $2$
Conductor $936$
Sign $-0.998 - 0.0540i$
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  + (0.526 + 1.31i)2-s + (−1.44 + 1.38i)4-s − 0.112·5-s + (0.0378 + 0.0218i)7-s + (−2.57 − 1.16i)8-s + (−0.0592 − 0.147i)10-s + (3.10 + 5.37i)11-s + (−1.35 − 3.34i)13-s + (−0.00874 + 0.0612i)14-s + (0.176 − 3.99i)16-s + (−1.70 + 2.95i)17-s + (−3.27 + 5.66i)19-s + (0.162 − 0.155i)20-s + (−5.41 + 6.90i)22-s + (2.27 + 3.93i)23-s + ⋯
L(s)  = 1  + (0.372 + 0.928i)2-s + (−0.722 + 0.691i)4-s − 0.0502·5-s + (0.0143 + 0.00826i)7-s + (−0.910 − 0.412i)8-s + (−0.0187 − 0.0466i)10-s + (0.935 + 1.62i)11-s + (−0.375 − 0.926i)13-s + (−0.00233 + 0.0163i)14-s + (0.0440 − 0.999i)16-s + (−0.413 + 0.716i)17-s + (−0.750 + 1.30i)19-s + (0.0363 − 0.0347i)20-s + (−1.15 + 1.47i)22-s + (0.474 + 0.821i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.998 - 0.0540i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.998 - 0.0540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0334001 + 1.23543i\)
\(L(\frac12)\) \(\approx\) \(0.0334001 + 1.23543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.526 - 1.31i)T \)
3 \( 1 \)
13 \( 1 + (1.35 + 3.34i)T \)
good5 \( 1 + 0.112T + 5T^{2} \)
7 \( 1 + (-0.0378 - 0.0218i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.10 - 5.37i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.70 - 2.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.27 - 5.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.27 - 3.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.22 - 3.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.66iT - 31T^{2} \)
37 \( 1 + (-5.07 - 8.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.01 - 1.16i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.02 + 4.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.65iT - 47T^{2} \)
53 \( 1 - 11.2iT - 53T^{2} \)
59 \( 1 + (-1.11 + 1.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.56 - 4.94i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.26 - 3.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.64 + 1.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.71iT - 73T^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 + (3.35 - 1.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.971 + 0.560i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.06001884593948666176325925161, −9.692542052848500258333274748796, −8.571740928238894508754122543741, −7.80473254915528851117620449052, −7.08099430149904829772829106018, −6.20679876489997571162369266185, −5.34415138834829531609497929808, −4.29652924176114695416954364679, −3.60622326780737646023680164793, −1.91652109938828937227987338171, 0.50527513267550672768874353832, 2.05035912757982380974261388965, 3.17487928773878025433446273811, 4.14969671924086560988049454197, 4.99228750680950526863390424007, 6.14804112726493981717816687676, 6.82921805862686185067108911303, 8.320788899949147386545516858437, 9.095780676725868770437409628441, 9.521819481099713677808343067234

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