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Properties

Label 2-943-943.620-c0-0-0
Degree $2$
Conductor $943$
Sign $-0.326 + 0.945i$
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 + (−1.41 − 1.41i)3-s + (−0.722 + 2.22i)4-s + (−0.570 + 3.60i)6-s + (2.32 − 0.755i)8-s + 2.98i·9-s + (4.16 − 2.12i)12-s + (1.65 + 0.262i)13-s + (−1.72 − 1.25i)16-s + (4.41 − 3.21i)18-s + (−0.809 + 0.587i)23-s + (−4.35 − 2.21i)24-s + (0.809 + 0.587i)25-s + (−1.39 − 2.73i)26-s + (2.80 − 2.80i)27-s + ⋯
L(s)  = 1  + (−1.07 − 1.47i)2-s + (−1.41 − 1.41i)3-s + (−0.722 + 2.22i)4-s + (−0.570 + 3.60i)6-s + (2.32 − 0.755i)8-s + 2.98i·9-s + (4.16 − 2.12i)12-s + (1.65 + 0.262i)13-s + (−1.72 − 1.25i)16-s + (4.41 − 3.21i)18-s + (−0.809 + 0.587i)23-s + (−4.35 − 2.21i)24-s + (0.809 + 0.587i)25-s + (−1.39 − 2.73i)26-s + (2.80 − 2.80i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(943\)    =    \(23 \cdot 41\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(0.470618\)
Root analytic conductor: \(0.686016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{943} (620, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 943,\ (\ :0),\ -0.326 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3463069786\)
\(L(\frac12)\) \(\approx\) \(0.3463069786\)
\(L(1)\) 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 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
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 \( 1 + (-1.65 - 0.262i)T + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.0932 - 0.0475i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (0.196 - 1.24i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (-0.847 + 1.66i)T + (-0.587 - 0.809i)T^{2} \)
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

−10.47280445226074728520078968423, −9.275070262543896494215913711822, −8.308570250876335975380960157127, −7.72366673851043863281428627636, −6.70755034505644255098978040585, −5.88508095362141900405523200508, −4.59332366405262596599689773986, −3.15338001997612970744843883878, −1.76753462718111631334388473698, −1.11246441615025548947746899421, 0.75855776052480419307337607101, 3.81421810932726733025740033072, 4.72272280189319622246015267961, 5.69451588014545128798678292854, 6.15547440133848550108766614616, 6.81036005996066294758629938924, 8.196309824033087672318503816072, 8.843447061621586439443426215449, 9.685518382108228156054731382770, 10.26941745841035307779573398035

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