Properties

Label 2-943-943.367-c0-0-0
Degree $2$
Conductor $943$
Sign $-0.451 - 0.892i$
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  + (−1.27 + 0.413i)2-s + (1.09 + 1.09i)3-s + (0.639 − 0.464i)4-s + (−1.85 − 0.944i)6-s + (0.164 − 0.226i)8-s + 1.41i·9-s + (1.21 + 0.192i)12-s + (−0.325 + 0.638i)13-s + (−0.360 + 1.10i)16-s + (−0.585 − 1.80i)18-s + (0.309 + 0.951i)23-s + (0.429 − 0.0680i)24-s + (−0.309 + 0.951i)25-s + (0.150 − 0.947i)26-s + (−0.457 + 0.457i)27-s + ⋯
L(s)  = 1  + (−1.27 + 0.413i)2-s + (1.09 + 1.09i)3-s + (0.639 − 0.464i)4-s + (−1.85 − 0.944i)6-s + (0.164 − 0.226i)8-s + 1.41i·9-s + (1.21 + 0.192i)12-s + (−0.325 + 0.638i)13-s + (−0.360 + 1.10i)16-s + (−0.585 − 1.80i)18-s + (0.309 + 0.951i)23-s + (0.429 − 0.0680i)24-s + (−0.309 + 0.951i)25-s + (0.150 − 0.947i)26-s + (−0.457 + 0.457i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(943\)    =    \(23 \cdot 41\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(0.470618\)
Root analytic conductor: \(0.686016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{943} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 943,\ (\ :0),\ -0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7250202520\)
\(L(\frac12)\) \(\approx\) \(0.7250202520\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (1.27 - 0.413i)T + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (-1.09 - 1.09i)T + iT^{2} \)
5 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
11 \( 1 + (0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.325 - 0.638i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (1.24 + 0.196i)T + (0.951 + 0.309i)T^{2} \)
31 \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (0.970 + 0.494i)T + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.0163 + 0.103i)T + (-0.951 + 0.309i)T^{2} \)
73 \( 1 + 0.813iT - T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.587 + 0.809i)T^{2} \)
97 \( 1 + (0.951 + 0.309i)T^{2} \)
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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

−10.08855692253400491694128282716, −9.425135383110892426527583083258, −9.066420909680423845277469526297, −8.226908582434092367795711024339, −7.53942026361073205951675021125, −6.64851291175929894073077799241, −5.20572720753149043257091143722, −4.14083576118655955027482463183, −3.28110629415995091733490812647, −1.84231592711039182908169170396, 1.00765547252638300073012316760, 2.26550828896855368535095418521, 2.89000792570922135211314014994, 4.49065968049740818050439830864, 6.00630687896542402950272865885, 7.07533574877586009598034822496, 7.86947313193422329317731078208, 8.228452966697589253511175456805, 9.064446893067474349427227144408, 9.766411779110085824439079780862

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