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Properties

Label 2-837-93.35-c0-0-0
Degree $2$
Conductor $837$
Sign $0.938 + 0.344i$
Analytic cond. $0.417717$
Root an. cond. $0.646310$
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  + (−1.53 + 0.5i)2-s + (1.30 − 0.951i)4-s − i·5-s + (−0.587 + 0.809i)8-s + (0.5 + 1.53i)10-s + (0.587 + 0.809i)11-s + (−0.363 + 0.5i)17-s + (−0.309 − 0.951i)19-s + (−0.951 − 1.30i)20-s + (−1.30 − 0.951i)22-s + (0.951 − 1.30i)23-s + (0.951 − 0.309i)29-s + (0.809 − 0.587i)31-s − 0.999i·32-s + (0.309 − 0.951i)34-s + ⋯
L(s)  = 1  + (−1.53 + 0.5i)2-s + (1.30 − 0.951i)4-s − i·5-s + (−0.587 + 0.809i)8-s + (0.5 + 1.53i)10-s + (0.587 + 0.809i)11-s + (−0.363 + 0.5i)17-s + (−0.309 − 0.951i)19-s + (−0.951 − 1.30i)20-s + (−1.30 − 0.951i)22-s + (0.951 − 1.30i)23-s + (0.951 − 0.309i)29-s + (0.809 − 0.587i)31-s − 0.999i·32-s + (0.309 − 0.951i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.938 + 0.344i$
Analytic conductor: \(0.417717\)
Root analytic conductor: \(0.646310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 837,\ (\ :0),\ 0.938 + 0.344i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4810287860\)
\(L(\frac12)\) \(\approx\) \(0.4810287860\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
5 \( 1 + iT - T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
11 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−10.16344476150271049422952497460, −9.210677750325588935569590805143, −8.736228150024339442678019599382, −8.162291924898168881094104716282, −6.92783502635423612952684567053, −6.57085213781958957841142290986, −5.10693086777086566740327038132, −4.22593430562870765072122832016, −2.25593746463127300379149121017, −0.942555025503386208010553402199, 1.36891080132298010366522084950, 2.74324710513884457180376487915, 3.56107748283399929516708597634, 5.28400491617995346646719818258, 6.69545039031828038342613438271, 7.04692431261200736925476386367, 8.281451746355369796335694195895, 8.722683144946593387391544388113, 9.796632394778257042420732424229, 10.27906049425518378376811984485

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