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Properties

Label 2-847-847.419-c0-0-0
Degree $2$
Conductor $847$
Sign $0.318 + 0.947i$
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  + (−0.797 − 0.234i)2-s + (−0.260 − 0.167i)4-s + (0.415 − 0.909i)7-s + (0.712 + 0.822i)8-s + 9-s + (−0.959 − 0.281i)11-s + (−0.544 + 0.627i)14-s + (−0.246 − 0.540i)16-s + (−0.797 − 0.234i)18-s + (0.698 + 0.449i)22-s + (0.345 + 0.755i)23-s + (−0.142 − 0.989i)25-s + (−0.260 + 0.167i)28-s + (0.186 − 1.29i)29-s + (−0.0845 − 0.588i)32-s + ⋯
L(s)  = 1  + (−0.797 − 0.234i)2-s + (−0.260 − 0.167i)4-s + (0.415 − 0.909i)7-s + (0.712 + 0.822i)8-s + 9-s + (−0.959 − 0.281i)11-s + (−0.544 + 0.627i)14-s + (−0.246 − 0.540i)16-s + (−0.797 − 0.234i)18-s + (0.698 + 0.449i)22-s + (0.345 + 0.755i)23-s + (−0.142 − 0.989i)25-s + (−0.260 + 0.167i)28-s + (0.186 − 1.29i)29-s + (−0.0845 − 0.588i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.318 + 0.947i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.318 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6212876836\)
\(L(\frac12)\) \(\approx\) \(0.6212876836\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.415 + 0.909i)T \)
11 \( 1 + (0.959 + 0.281i)T \)
good2 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (-0.415 - 0.909i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.959 + 0.281i)T^{2} \)
23 \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.415 + 0.909i)T^{2} \)
37 \( 1 + (-1.68 + 1.08i)T + (0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.841 + 0.540i)T^{2} \)
53 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.841 + 0.540i)T^{2} \)
67 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (0.959 - 0.281i)T^{2} \)
97 \( 1 + (0.142 - 0.989i)T^{2} \)
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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

−10.04092039375731530588258743546, −9.711142126674094420440023106584, −8.481272703051890502315705472818, −7.81209705600666200266351037642, −7.17423816539076282912359648369, −5.80748073911341480330681261239, −4.73807485747163504579632230383, −4.01356044806551444821567780187, −2.28901676368649776640292863320, −0.942693466188934649593596428753, 1.55762667507813373628196350623, 3.03128453526091516159468638464, 4.47877496555149045634428215980, 5.12662721980053131492255348875, 6.47335304794781555298192859990, 7.43894020768550341115307961136, 8.049159022184396933403386432053, 8.876452577049582604285208530558, 9.605376992353316182686998390515, 10.32245508340206003856039093160

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