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Properties

Label 2-847-11.5-c1-0-51
Degree $2$
Conductor $847$
Sign $0.598 - 0.801i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.651 − 2.00i)2-s + (−1.37 − 0.998i)3-s + (−1.98 + 1.43i)4-s + (0.152 − 0.468i)5-s + (−1.10 + 3.40i)6-s + (0.809 − 0.587i)7-s + (0.767 + 0.557i)8-s + (−0.0355 − 0.109i)9-s − 1.03·10-s + 4.16·12-s + (−1.63 − 5.04i)13-s + (−1.70 − 1.23i)14-s + (−0.676 + 0.491i)15-s + (−0.895 + 2.75i)16-s + (−0.938 + 2.88i)17-s + (−0.196 + 0.142i)18-s + ⋯
L(s)  = 1  + (−0.460 − 1.41i)2-s + (−0.793 − 0.576i)3-s + (−0.990 + 0.719i)4-s + (0.0680 − 0.209i)5-s + (−0.451 + 1.39i)6-s + (0.305 − 0.222i)7-s + (0.271 + 0.197i)8-s + (−0.0118 − 0.0364i)9-s − 0.328·10-s + 1.20·12-s + (−0.454 − 1.39i)13-s + (−0.456 − 0.331i)14-s + (−0.174 + 0.126i)15-s + (−0.223 + 0.689i)16-s + (−0.227 + 0.700i)17-s + (−0.0462 + 0.0336i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.598 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.241457 + 0.121049i\)
\(L(\frac12)\) \(\approx\) \(0.241457 + 0.121049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.651 + 2.00i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.37 + 0.998i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.152 + 0.468i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.63 + 5.04i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.938 - 2.88i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.77 + 2.74i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.63T + 23T^{2} \)
29 \( 1 + (-5.60 + 4.06i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.391 + 1.20i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.79 - 6.39i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.16 + 0.848i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.88T + 43T^{2} \)
47 \( 1 + (-7.08 - 5.14i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.05 - 6.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.76 + 4.91i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.29 + 13.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.70T + 67T^{2} \)
71 \( 1 + (-1.83 + 5.65i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.05 - 2.21i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.72 - 8.37i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.42 - 10.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (1.95 + 6.00i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.828532346275564444470286265423, −8.730348565870812409713898665587, −8.062617551250303093863625000101, −6.81339213899484271301966065959, −5.96719996136557663938645581417, −4.86480654720906028236868983342, −3.66712644608867725728777157295, −2.49737840449041568719693900388, −1.27979580949528758658234319984, −0.17939827296956414323675162359, 2.27690440474876268896079549653, 4.22360878269078044946071705609, 5.00753273295915147574238487304, 5.79701995167384086800571048254, 6.66224989801319935039179607315, 7.26842658080049289102913058064, 8.455711400566745795156645059687, 8.917918151005275934464495558227, 10.04780813714698037269405766556, 10.59792437724307973520342896265

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