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Properties

Label 2-837-93.8-c0-0-1
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: $\chi_{837} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 837,\ (\ :0),\ 0.938 - 0.344i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.080737575\)
\(L(\frac12)\) \(\approx\) \(2.080737575\)
\(L(1)\) 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

−10.49421117190898751285211190116, −9.708166493861636962576476096972, −8.448662552919655886817967134683, −7.81514760189027832879568808488, −6.69211494783427294558066066122, −5.90347199890078154799995545721, −4.98089685368059235320753326829, −4.44543020689502498327056005038, −3.41552451741835423546073239216, −1.98814132334355610867878810614, 2.15490658962827944879379852297, 3.11762364962037886886323854241, 3.75973137285419100937261065854, 5.05218867183817940890261641052, 5.72969043272072271767248227865, 6.64795302659727716513528156275, 7.48276410328315929881981725522, 8.668985446003636159479618744583, 9.920918866699110604725234582858, 10.73082886215111154726498754038

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