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Properties

Label 2-847-847.412-c0-0-0
Degree $2$
Conductor $847$
Sign $0.999 + 0.0311i$
Analytic cond. $0.422708$
Root an. cond. $0.650160$
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  + (−1.73 − 0.733i)2-s + (1.77 + 1.82i)4-s + (0.941 − 0.336i)7-s + (−1.05 − 2.71i)8-s + (0.309 + 0.951i)9-s + (0.516 + 0.856i)11-s + (−1.88 − 0.107i)14-s + (−0.0841 + 2.94i)16-s + (0.161 − 1.87i)18-s + (−0.268 − 1.86i)22-s + (−1.17 + 0.344i)23-s + (−0.870 + 0.491i)25-s + (2.28 + 1.12i)28-s + (0.442 + 0.250i)29-s + (1.09 − 2.39i)32-s + ⋯
L(s)  = 1  + (−1.73 − 0.733i)2-s + (1.77 + 1.82i)4-s + (0.941 − 0.336i)7-s + (−1.05 − 2.71i)8-s + (0.309 + 0.951i)9-s + (0.516 + 0.856i)11-s + (−1.88 − 0.107i)14-s + (−0.0841 + 2.94i)16-s + (0.161 − 1.87i)18-s + (−0.268 − 1.86i)22-s + (−1.17 + 0.344i)23-s + (−0.870 + 0.491i)25-s + (2.28 + 1.12i)28-s + (0.442 + 0.250i)29-s + (1.09 − 2.39i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.999 + 0.0311i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.999 + 0.0311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5092243069\)
\(L(\frac12)\) \(\approx\) \(0.5092243069\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.941 + 0.336i)T \)
11 \( 1 + (-0.516 - 0.856i)T \)
good2 \( 1 + (1.73 + 0.733i)T + (0.696 + 0.717i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.870 - 0.491i)T^{2} \)
13 \( 1 + (-0.610 + 0.791i)T^{2} \)
17 \( 1 + (-0.0855 - 0.996i)T^{2} \)
19 \( 1 + (0.921 + 0.389i)T^{2} \)
23 \( 1 + (1.17 - 0.344i)T + (0.841 - 0.540i)T^{2} \)
29 \( 1 + (-0.442 - 0.250i)T + (0.516 + 0.856i)T^{2} \)
31 \( 1 + (-0.941 + 0.336i)T^{2} \)
37 \( 1 + (-0.755 - 1.43i)T + (-0.564 + 0.825i)T^{2} \)
41 \( 1 + (-0.897 + 0.441i)T^{2} \)
43 \( 1 + (0.0480 + 0.0308i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (0.985 - 0.170i)T^{2} \)
53 \( 1 + (0.0567 + 1.98i)T + (-0.998 + 0.0570i)T^{2} \)
59 \( 1 + (-0.897 - 0.441i)T^{2} \)
61 \( 1 + (-0.696 + 0.717i)T^{2} \)
67 \( 1 + (-0.964 + 1.11i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-1.78 + 0.204i)T + (0.974 - 0.226i)T^{2} \)
73 \( 1 + (0.254 + 0.967i)T^{2} \)
79 \( 1 + (1.52 + 1.24i)T + (0.198 + 0.980i)T^{2} \)
83 \( 1 + (0.362 + 0.931i)T^{2} \)
89 \( 1 + (0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.870 + 0.491i)T^{2} \)
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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

−10.01599518606685944472324624486, −9.928649200786647119770728235147, −8.709987686153630710823351633674, −7.930192486781206499506771142904, −7.52736540982630706828126243451, −6.53179108740167928166646007314, −4.89673258956946771253741565998, −3.75383085630929343742733986675, −2.19826142634991733958628063746, −1.52889947445394165962045150066, 1.02144961135732549106316415483, 2.30283782138418730492574722462, 4.11608026331327297968985671074, 5.73419422576270743337468415654, 6.20012094995187701571911656669, 7.21866645680026088203071344868, 8.072839888528469936511678818781, 8.634907423602557257115630816609, 9.390253639065603833197655279471, 10.08414677051691812135786461626

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