Properties

Label 2-847-847.104-c0-0-0
Degree $2$
Conductor $847$
Sign $0.560 - 0.828i$
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  + (0.714 − 0.123i)2-s + (−0.445 + 0.159i)4-s + (−0.362 + 0.931i)7-s + (−0.930 + 0.525i)8-s + (0.309 + 0.951i)9-s + (0.897 + 0.441i)11-s + (−0.144 + 0.711i)14-s + (−0.234 + 0.191i)16-s + (0.338 + 0.641i)18-s + (0.696 + 0.204i)22-s + (−1.67 − 1.07i)23-s + (0.974 − 0.226i)25-s + (0.0135 − 0.473i)28-s + (1.93 + 0.450i)29-s + (0.556 − 0.642i)32-s + ⋯
L(s)  = 1  + (0.714 − 0.123i)2-s + (−0.445 + 0.159i)4-s + (−0.362 + 0.931i)7-s + (−0.930 + 0.525i)8-s + (0.309 + 0.951i)9-s + (0.897 + 0.441i)11-s + (−0.144 + 0.711i)14-s + (−0.234 + 0.191i)16-s + (0.338 + 0.641i)18-s + (0.696 + 0.204i)22-s + (−1.67 − 1.07i)23-s + (0.974 − 0.226i)25-s + (0.0135 − 0.473i)28-s + (1.93 + 0.450i)29-s + (0.556 − 0.642i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.560 - 0.828i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.560 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.153563046\)
\(L(\frac12)\) \(\approx\) \(1.153563046\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.362 - 0.931i)T \)
11 \( 1 + (-0.897 - 0.441i)T \)
good2 \( 1 + (-0.714 + 0.123i)T + (0.941 - 0.336i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.974 + 0.226i)T^{2} \)
13 \( 1 + (0.998 - 0.0570i)T^{2} \)
17 \( 1 + (0.466 - 0.884i)T^{2} \)
19 \( 1 + (0.985 - 0.170i)T^{2} \)
23 \( 1 + (1.67 + 1.07i)T + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (-1.93 - 0.450i)T + (0.897 + 0.441i)T^{2} \)
31 \( 1 + (0.362 - 0.931i)T^{2} \)
37 \( 1 + (0.988 - 1.28i)T + (-0.254 - 0.967i)T^{2} \)
41 \( 1 + (0.0285 + 0.999i)T^{2} \)
43 \( 1 + (-0.643 + 1.40i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (0.564 + 0.825i)T^{2} \)
53 \( 1 + (1.42 + 1.16i)T + (0.198 + 0.980i)T^{2} \)
59 \( 1 + (0.0285 - 0.999i)T^{2} \)
61 \( 1 + (-0.941 - 0.336i)T^{2} \)
67 \( 1 + (0.146 - 1.02i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (-0.0526 + 0.0222i)T + (0.696 - 0.717i)T^{2} \)
73 \( 1 + (-0.993 + 0.113i)T^{2} \)
79 \( 1 + (-0.831 + 0.763i)T + (0.0855 - 0.996i)T^{2} \)
83 \( 1 + (0.870 - 0.491i)T^{2} \)
89 \( 1 + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (-0.974 - 0.226i)T^{2} \)
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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

−10.43591783198536962541173487064, −9.750110129526442859784071849036, −8.631636110619277519488139341728, −8.329813305261806828115345367105, −6.85172925828106887385416167991, −6.07043687096265930445806504300, −4.98035998895321802660086988860, −4.40204784764084286299842005404, −3.18481119960163735097739016727, −2.11723633429636965931825703519, 1.04524048779261274252331974742, 3.27647339124353760982223109591, 3.92153719142469480771973159200, 4.70650848273928899187585027926, 6.10154290094316887099683367307, 6.44370565153869072836892726012, 7.55159345517088585453932026893, 8.741651085655062156907394306985, 9.507438123593423031410327752963, 10.08485731908743686038418141484

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