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Properties

Label 2-847-847.510-c0-0-0
Degree $2$
Conductor $847$
Sign $0.279 + 0.960i$
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  + (−0.0556 + 0.0129i)2-s + (−0.894 + 0.439i)4-s + (−0.0285 − 0.999i)7-s + (0.0882 − 0.0721i)8-s + (−0.809 − 0.587i)9-s + (0.0855 − 0.996i)11-s + (0.0145 + 0.0552i)14-s + (0.604 − 0.784i)16-s + (0.0526 + 0.0222i)18-s + (0.00812 + 0.0565i)22-s + (1.08 − 0.318i)23-s + (−0.736 − 0.676i)25-s + (0.465 + 0.881i)28-s + (0.534 − 0.490i)29-s + (−0.0708 + 0.155i)32-s + ⋯
L(s)  = 1  + (−0.0556 + 0.0129i)2-s + (−0.894 + 0.439i)4-s + (−0.0285 − 0.999i)7-s + (0.0882 − 0.0721i)8-s + (−0.809 − 0.587i)9-s + (0.0855 − 0.996i)11-s + (0.0145 + 0.0552i)14-s + (0.604 − 0.784i)16-s + (0.0526 + 0.0222i)18-s + (0.00812 + 0.0565i)22-s + (1.08 − 0.318i)23-s + (−0.736 − 0.676i)25-s + (0.465 + 0.881i)28-s + (0.534 − 0.490i)29-s + (−0.0708 + 0.155i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (510, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.279 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6544311570\)
\(L(\frac12)\) \(\approx\) \(0.6544311570\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.0285 + 0.999i)T \)
11 \( 1 + (-0.0855 + 0.996i)T \)
good2 \( 1 + (0.0556 - 0.0129i)T + (0.897 - 0.441i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.736 + 0.676i)T^{2} \)
13 \( 1 + (0.564 + 0.825i)T^{2} \)
17 \( 1 + (0.921 - 0.389i)T^{2} \)
19 \( 1 + (-0.974 + 0.226i)T^{2} \)
23 \( 1 + (-1.08 + 0.318i)T + (0.841 - 0.540i)T^{2} \)
29 \( 1 + (-0.534 + 0.490i)T + (0.0855 - 0.996i)T^{2} \)
31 \( 1 + (0.0285 + 0.999i)T^{2} \)
37 \( 1 + (0.608 - 0.105i)T + (0.941 - 0.336i)T^{2} \)
41 \( 1 + (0.466 - 0.884i)T^{2} \)
43 \( 1 + (-1.02 - 0.660i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.696 + 0.717i)T^{2} \)
53 \( 1 + (1.06 + 1.37i)T + (-0.254 + 0.967i)T^{2} \)
59 \( 1 + (0.466 + 0.884i)T^{2} \)
61 \( 1 + (-0.897 - 0.441i)T^{2} \)
67 \( 1 + (0.260 - 0.300i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.812 - 0.458i)T + (0.516 + 0.856i)T^{2} \)
73 \( 1 + (0.362 - 0.931i)T^{2} \)
79 \( 1 + (1.39 + 0.0795i)T + (0.993 + 0.113i)T^{2} \)
83 \( 1 + (-0.774 + 0.633i)T^{2} \)
89 \( 1 + (0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.736 - 0.676i)T^{2} \)
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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

−10.12952868087663233546464406018, −9.294544845934055348267538556173, −8.512034386892336421804039734551, −7.906277359844952661370780771153, −6.79075403746797049488838372484, −5.85163825780081700809715111745, −4.73713341563802339769413814367, −3.78505839738951363248126110450, −2.99871856817718547351305961660, −0.71405774997615329338018677057, 1.81602140113318988470016200576, 3.10104603698955017542373149437, 4.51645477830632643096584297380, 5.28738451206361919314060684715, 5.94084105954440905423239509779, 7.23615817617012506052391918685, 8.239148272758381706203417750753, 9.020482983730077194778281862365, 9.509498631239948750721143552314, 10.51149322610996517967727662554

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