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Properties

Label 2-847-847.181-c0-0-0
Degree $2$
Conductor $847$
Sign $0.999 - 0.0103i$
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  + (−0.0113 − 0.397i)2-s + (0.840 − 0.0480i)4-s + (0.198 + 0.980i)7-s + (−0.0626 − 0.729i)8-s + (0.309 + 0.951i)9-s + (−0.564 + 0.825i)11-s + (0.386 − 0.0899i)14-s + (0.548 − 0.0628i)16-s + (0.374 − 0.133i)18-s + (0.334 + 0.214i)22-s + (−0.723 − 1.58i)23-s + (−0.466 + 0.884i)25-s + (0.214 + 0.814i)28-s + (−0.481 − 0.913i)29-s + (−0.135 − 0.941i)32-s + ⋯
L(s)  = 1  + (−0.0113 − 0.397i)2-s + (0.840 − 0.0480i)4-s + (0.198 + 0.980i)7-s + (−0.0626 − 0.729i)8-s + (0.309 + 0.951i)9-s + (−0.564 + 0.825i)11-s + (0.386 − 0.0899i)14-s + (0.548 − 0.0628i)16-s + (0.374 − 0.133i)18-s + (0.334 + 0.214i)22-s + (−0.723 − 1.58i)23-s + (−0.466 + 0.884i)25-s + (0.214 + 0.814i)28-s + (−0.481 − 0.913i)29-s + (−0.135 − 0.941i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.999 - 0.0103i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.999 - 0.0103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.189932029\)
\(L(\frac12)\) \(\approx\) \(1.189932029\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.198 - 0.980i)T \)
11 \( 1 + (0.564 - 0.825i)T \)
good2 \( 1 + (0.0113 + 0.397i)T + (-0.998 + 0.0570i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.466 - 0.884i)T^{2} \)
13 \( 1 + (0.870 + 0.491i)T^{2} \)
17 \( 1 + (-0.941 - 0.336i)T^{2} \)
19 \( 1 + (0.0285 + 0.999i)T^{2} \)
23 \( 1 + (0.723 + 1.58i)T + (-0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.481 + 0.913i)T + (-0.564 + 0.825i)T^{2} \)
31 \( 1 + (-0.198 - 0.980i)T^{2} \)
37 \( 1 + (-0.587 + 1.50i)T + (-0.736 - 0.676i)T^{2} \)
41 \( 1 + (0.254 - 0.967i)T^{2} \)
43 \( 1 + (1.30 + 1.50i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.774 - 0.633i)T^{2} \)
53 \( 1 + (-1.78 - 0.204i)T + (0.974 + 0.226i)T^{2} \)
59 \( 1 + (0.254 + 0.967i)T^{2} \)
61 \( 1 + (0.998 + 0.0570i)T^{2} \)
67 \( 1 + (-1.89 - 0.555i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.456 + 0.224i)T + (0.610 + 0.791i)T^{2} \)
73 \( 1 + (-0.516 - 0.856i)T^{2} \)
79 \( 1 + (1.42 + 0.602i)T + (0.696 + 0.717i)T^{2} \)
83 \( 1 + (-0.0855 - 0.996i)T^{2} \)
89 \( 1 + (0.959 - 0.281i)T^{2} \)
97 \( 1 + (0.466 + 0.884i)T^{2} \)
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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

−10.37732127671830829592286298487, −9.863751640143898169823990669768, −8.693747108019045334803063619507, −7.77026387829515239497592704866, −7.09412229060336564598390331540, −5.96060326753064127102799952426, −5.16504533257879074038057027001, −3.97738849090124152789394174865, −2.41030344467339372099917799571, −2.05956635212086491592144161902, 1.43143259449153898488144398494, 3.03922596162035824411643682415, 3.92620955128245409072048017470, 5.28919274882472013189902489610, 6.22525602782840427745302285840, 6.93060610359435978788872681625, 7.78486166914986909951615206695, 8.409075817118772343960619994895, 9.742225559758443321001182768739, 10.32745862095373109723480712754

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