Properties

Label 2-943-943.620-c0-0-2
Degree $2$
Conductor $943$
Sign $0.981 - 0.190i$
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  + (0.122 + 0.169i)2-s + (0.770 + 0.770i)3-s + (0.295 − 0.909i)4-s + (−0.0356 + 0.224i)6-s + (0.388 − 0.126i)8-s + 0.186i·9-s + (0.928 − 0.472i)12-s + (0.103 + 0.0163i)13-s + (−0.704 − 0.511i)16-s + (−0.0315 + 0.0229i)18-s + (−0.809 + 0.587i)23-s + (0.396 + 0.202i)24-s + (0.809 + 0.587i)25-s + (0.00993 + 0.0194i)26-s + (0.626 − 0.626i)27-s + ⋯
L(s)  = 1  + (0.122 + 0.169i)2-s + (0.770 + 0.770i)3-s + (0.295 − 0.909i)4-s + (−0.0356 + 0.224i)6-s + (0.388 − 0.126i)8-s + 0.186i·9-s + (0.928 − 0.472i)12-s + (0.103 + 0.0163i)13-s + (−0.704 − 0.511i)16-s + (−0.0315 + 0.0229i)18-s + (−0.809 + 0.587i)23-s + (0.396 + 0.202i)24-s + (0.809 + 0.587i)25-s + (0.00993 + 0.0194i)26-s + (0.626 − 0.626i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(943\)    =    \(23 \cdot 41\)
Sign: $0.981 - 0.190i$
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: \((2,\ 943,\ (\ :0),\ 0.981 - 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.441807599\)
\(L(\frac12)\) \(\approx\) \(1.441807599\)
\(L(1)\) 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 \( 1 + (0.104 + 0.994i)T \)
good2 \( 1 + (-0.122 - 0.169i)T + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.770 - 0.770i)T + iT^{2} \)
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 \( 1 + (-0.103 - 0.0163i)T + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (1.49 - 0.761i)T + (0.587 - 0.809i)T^{2} \)
31 \( 1 + (-0.128 - 0.395i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.309 - 0.951i)T^{2} \)
47 \( 1 + (0.112 - 0.707i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (0.705 - 1.38i)T + (-0.587 - 0.809i)T^{2} \)
73 \( 1 + 1.48iT - T^{2} \)
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

−10.21080877286324565183579193813, −9.380647167009585430071983908632, −8.934603913982091623358112769062, −7.76255583233376436597265894084, −6.85672364416859568704786712265, −5.87461978749524781712155621327, −5.01659908301187764365989226746, −4.00664192571358988147951389327, −3.01914642861865995209057695045, −1.65512285823835893651206760768, 1.86765192223819035797346627825, 2.69811017509965093664790945403, 3.68662332147519606169322804005, 4.74702191694468367763307408364, 6.19397249008780479112416962553, 7.03827276911629233861214934323, 7.86000461825369188206182958914, 8.284071464069863221998786216627, 9.159506489437112664199459871876, 10.26938621849042786673257482348

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