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Properties

Label 2-847-847.279-c0-0-0
Degree $2$
Conductor $847$
Sign $-0.860 + 0.509i$
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  + (−0.374 − 0.962i)2-s + (−0.0491 + 0.0450i)4-s + (0.516 − 0.856i)7-s + (−0.865 − 0.425i)8-s + (−0.809 − 0.587i)9-s + (−0.998 + 0.0570i)11-s + (−1.01 − 0.176i)14-s + (−0.0908 + 1.05i)16-s + (−0.262 + 0.998i)18-s + (0.429 + 0.939i)22-s + (1.20 − 1.39i)23-s + (−0.0285 + 0.999i)25-s + (0.0132 + 0.0653i)28-s + (−0.0398 − 1.39i)29-s + (0.127 − 0.0375i)32-s + ⋯
L(s)  = 1  + (−0.374 − 0.962i)2-s + (−0.0491 + 0.0450i)4-s + (0.516 − 0.856i)7-s + (−0.865 − 0.425i)8-s + (−0.809 − 0.587i)9-s + (−0.998 + 0.0570i)11-s + (−1.01 − 0.176i)14-s + (−0.0908 + 1.05i)16-s + (−0.262 + 0.998i)18-s + (0.429 + 0.939i)22-s + (1.20 − 1.39i)23-s + (−0.0285 + 0.999i)25-s + (0.0132 + 0.0653i)28-s + (−0.0398 − 1.39i)29-s + (0.127 − 0.0375i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ -0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7392076145\)
\(L(\frac12)\) \(\approx\) \(0.7392076145\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.516 + 0.856i)T \)
11 \( 1 + (0.998 - 0.0570i)T \)
good2 \( 1 + (0.374 + 0.962i)T + (-0.736 + 0.676i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.0285 - 0.999i)T^{2} \)
13 \( 1 + (0.921 + 0.389i)T^{2} \)
17 \( 1 + (0.254 + 0.967i)T^{2} \)
19 \( 1 + (0.362 + 0.931i)T^{2} \)
23 \( 1 + (-1.20 + 1.39i)T + (-0.142 - 0.989i)T^{2} \)
29 \( 1 + (0.0398 + 1.39i)T + (-0.998 + 0.0570i)T^{2} \)
31 \( 1 + (-0.516 + 0.856i)T^{2} \)
37 \( 1 + (-0.614 - 0.0704i)T + (0.974 + 0.226i)T^{2} \)
41 \( 1 + (-0.198 + 0.980i)T^{2} \)
43 \( 1 + (0.0243 - 0.169i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.870 - 0.491i)T^{2} \)
53 \( 1 + (-0.161 - 1.87i)T + (-0.985 + 0.170i)T^{2} \)
59 \( 1 + (-0.198 - 0.980i)T^{2} \)
61 \( 1 + (0.736 + 0.676i)T^{2} \)
67 \( 1 + (-1.02 - 0.660i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.374 + 0.133i)T + (0.774 - 0.633i)T^{2} \)
73 \( 1 + (-0.696 - 0.717i)T^{2} \)
79 \( 1 + (-0.812 + 1.54i)T + (-0.564 - 0.825i)T^{2} \)
83 \( 1 + (-0.897 - 0.441i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.0285 + 0.999i)T^{2} \)
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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

−10.23471652303831745409282480888, −9.387580029228334261911735691014, −8.555612279826766050781881883109, −7.63082965140743762467959348799, −6.60412747354981392843130192338, −5.63899143308142582774542099849, −4.47304656590461231411578994559, −3.25756887617435471602448008869, −2.37556297186642210985825734679, −0.829155862352993379557406266123, 2.27525863282665597092588294809, 3.17161177549952595557254127414, 5.17361218436841839519948174466, 5.38350465521368139613824736468, 6.50307286049344776836478616177, 7.50206231320496360227765255415, 8.195761670622949092213163986452, 8.708072217613988764790071784419, 9.606843800843382073181318381527, 10.89705857450494398383575425353

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