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Properties

Label 2-943-943.459-c0-0-2
Degree $2$
Conductor $943$
Sign $-0.771 + 0.635i$
Analytic cond. $0.470618$
Root an. cond. $0.686016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.07 − 1.47i)2-s + (0.0740 + 0.0740i)3-s + (−0.722 − 2.22i)4-s + (0.188 − 0.0299i)6-s + (−2.32 − 0.755i)8-s − 0.989i·9-s + (0.111 − 0.218i)12-s + (0.170 + 1.07i)13-s + (−1.72 + 1.25i)16-s + (−1.46 − 1.06i)18-s + (−0.809 − 0.587i)23-s + (−0.116 − 0.228i)24-s + (0.809 − 0.587i)25-s + (1.77 + 0.903i)26-s + (0.147 − 0.147i)27-s + ⋯
L(s)  = 1  + (1.07 − 1.47i)2-s + (0.0740 + 0.0740i)3-s + (−0.722 − 2.22i)4-s + (0.188 − 0.0299i)6-s + (−2.32 − 0.755i)8-s − 0.989i·9-s + (0.111 − 0.218i)12-s + (0.170 + 1.07i)13-s + (−1.72 + 1.25i)16-s + (−1.46 − 1.06i)18-s + (−0.809 − 0.587i)23-s + (−0.116 − 0.228i)24-s + (0.809 − 0.587i)25-s + (1.77 + 0.903i)26-s + (0.147 − 0.147i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(943\)    =    \(23 \cdot 41\)
Sign: $-0.771 + 0.635i$
Analytic conductor: \(0.470618\)
Root analytic conductor: \(0.686016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{943} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 943,\ (\ :0),\ -0.771 + 0.635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.641392779\)
\(L(\frac12)\) \(\approx\) \(1.641392779\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
good2 \( 1 + (-1.07 + 1.47i)T + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.0740 - 0.0740i)T + iT^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.951 - 0.309i)T^{2} \)
11 \( 1 + (-0.587 + 0.809i)T^{2} \)
13 \( 1 + (-0.170 - 1.07i)T + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.906 - 1.77i)T + (-0.587 - 0.809i)T^{2} \)
31 \( 1 + (0.459 - 1.41i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (-1.53 + 0.243i)T + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (0.638 - 0.325i)T + (0.587 - 0.809i)T^{2} \)
73 \( 1 - 0.415iT - T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.951 - 0.309i)T^{2} \)
97 \( 1 + (-0.587 - 0.809i)T^{2} \)
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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

−10.31301965436994546527634502479, −9.190213275913701192922933267058, −8.863646092330058662119275490241, −7.08604150756500676975675535578, −6.21540926794030483775973375603, −5.24700885102132402679280654358, −4.24546367565844646752449451716, −3.60926660909517479904738758303, −2.54830628156530267746049712925, −1.33407097231475687058636465056, 2.51359092718453813592786809355, 3.75390366548066767470078724825, 4.58595774128268751703791749328, 5.69416673131039220790486338368, 5.91421787832698169491262012079, 7.39188623499557249851149231522, 7.64824228953356553130970678537, 8.444222700162377801336451225233, 9.503166243101921745796523523722, 10.65213649021926812558680603577

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