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Properties

Label 2-847-847.587-c0-0-0
Degree $2$
Conductor $847$
Sign $0.299 + 0.954i$
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  + (−1.38 − 0.495i)2-s + (0.906 + 0.741i)4-s + (−0.736 + 0.676i)7-s + (−0.130 − 0.215i)8-s + (−0.809 − 0.587i)9-s + (0.610 − 0.791i)11-s + (1.35 − 0.573i)14-s + (−0.158 − 0.783i)16-s + (0.831 + 1.21i)18-s + (−1.23 + 0.796i)22-s + (0.825 − 1.80i)23-s + (0.897 + 0.441i)25-s + (−1.16 + 0.0668i)28-s + (1.74 − 0.859i)29-s + (−0.203 + 1.41i)32-s + ⋯
L(s)  = 1  + (−1.38 − 0.495i)2-s + (0.906 + 0.741i)4-s + (−0.736 + 0.676i)7-s + (−0.130 − 0.215i)8-s + (−0.809 − 0.587i)9-s + (0.610 − 0.791i)11-s + (1.35 − 0.573i)14-s + (−0.158 − 0.783i)16-s + (0.831 + 1.21i)18-s + (−1.23 + 0.796i)22-s + (0.825 − 1.80i)23-s + (0.897 + 0.441i)25-s + (−1.16 + 0.0668i)28-s + (1.74 − 0.859i)29-s + (−0.203 + 1.41i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.299 + 0.954i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.299 + 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4257523048\)
\(L(\frac12)\) \(\approx\) \(0.4257523048\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.736 - 0.676i)T \)
11 \( 1 + (-0.610 + 0.791i)T \)
good2 \( 1 + (1.38 + 0.495i)T + (0.774 + 0.633i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.897 - 0.441i)T^{2} \)
13 \( 1 + (-0.993 - 0.113i)T^{2} \)
17 \( 1 + (0.564 - 0.825i)T^{2} \)
19 \( 1 + (-0.941 - 0.336i)T^{2} \)
23 \( 1 + (-0.825 + 1.80i)T + (-0.654 - 0.755i)T^{2} \)
29 \( 1 + (-1.74 + 0.859i)T + (0.610 - 0.791i)T^{2} \)
31 \( 1 + (0.736 - 0.676i)T^{2} \)
37 \( 1 + (0.157 - 0.597i)T + (-0.870 - 0.491i)T^{2} \)
41 \( 1 + (0.998 + 0.0570i)T^{2} \)
43 \( 1 + (0.260 - 0.300i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.362 + 0.931i)T^{2} \)
53 \( 1 + (-0.276 + 1.36i)T + (-0.921 - 0.389i)T^{2} \)
59 \( 1 + (0.998 - 0.0570i)T^{2} \)
61 \( 1 + (-0.774 + 0.633i)T^{2} \)
67 \( 1 + (-0.895 + 0.262i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (1.39 + 1.43i)T + (-0.0285 + 0.999i)T^{2} \)
73 \( 1 + (-0.974 - 0.226i)T^{2} \)
79 \( 1 + (0.0620 + 0.722i)T + (-0.985 + 0.170i)T^{2} \)
83 \( 1 + (-0.516 - 0.856i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.897 + 0.441i)T^{2} \)
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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

−10.12584533075947348908154795008, −9.234637684001040817335104669027, −8.702735902040767104693339260397, −8.247701427366400337022166680758, −6.75046986467858623241403956943, −6.24125157676032147942860588512, −4.93119476009017065724196545027, −3.25593671046780892004667938433, −2.55652556525686257663084419122, −0.78339943409515863563198125819, 1.29064049538377355599395017470, 2.95117857765928195466509149142, 4.27836262368380151036182076446, 5.56009360472334843593021183532, 6.78341100732626063046507563268, 7.12562262589664245562890960953, 8.109745507266599332152387305799, 8.935680357694808082275024998509, 9.553515787651814583928677534176, 10.37241530713158890223302728982

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