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Properties

Label 2-847-77.76-c1-0-40
Degree $2$
Conductor $847$
Sign $0.762 + 0.647i$
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  + 0.502i·2-s − 2.88i·3-s + 1.74·4-s + 0.602i·5-s + 1.44·6-s + (2.34 + 1.22i)7-s + 1.88i·8-s − 5.29·9-s − 0.302·10-s − 5.03i·12-s + 2.34·13-s + (−0.618 + 1.17i)14-s + 1.73·15-s + 2.54·16-s + 1.14·17-s − 2.66i·18-s + ⋯
L(s)  = 1  + 0.355i·2-s − 1.66i·3-s + 0.873·4-s + 0.269i·5-s + 0.591·6-s + (0.885 + 0.464i)7-s + 0.666i·8-s − 1.76·9-s − 0.0957·10-s − 1.45i·12-s + 0.649·13-s + (−0.165 + 0.314i)14-s + 0.447·15-s + 0.636·16-s + 0.277·17-s − 0.627i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.762 + 0.647i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.762 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04614 - 0.751623i\)
\(L(\frac12)\) \(\approx\) \(2.04614 - 0.751623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.34 - 1.22i)T \)
11 \( 1 \)
good2 \( 1 - 0.502iT - 2T^{2} \)
3 \( 1 + 2.88iT - 3T^{2} \)
5 \( 1 - 0.602iT - 5T^{2} \)
13 \( 1 - 2.34T + 13T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
19 \( 1 - 5.07T + 19T^{2} \)
23 \( 1 + 3.85T + 23T^{2} \)
29 \( 1 - 2.15iT - 29T^{2} \)
31 \( 1 + 6.23iT - 31T^{2} \)
37 \( 1 + 7.41T + 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + 9.92iT - 47T^{2} \)
53 \( 1 - 6.12T + 53T^{2} \)
59 \( 1 - 4.70iT - 59T^{2} \)
61 \( 1 + 8.70T + 61T^{2} \)
67 \( 1 + 6.46T + 67T^{2} \)
71 \( 1 - 5.32T + 71T^{2} \)
73 \( 1 + 2.49T + 73T^{2} \)
79 \( 1 + 1.19iT - 79T^{2} \)
83 \( 1 - 7.90T + 83T^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
97 \( 1 - 5.82iT - 97T^{2} \)
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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

−10.32185268583067324001744491124, −8.765993117518594387222314648723, −8.122995885396157114451312083886, −7.42478928829738593427874335474, −6.80800437350581651584310918737, −5.95060162589701582244230798387, −5.25073936655772278516438031790, −3.25188061884290032238011949758, −2.15472147434407563467225615107, −1.34295013276977790576426801157, 1.44909659994180271175302988277, 3.08956063484243096861592498069, 3.81873052495154216326274142999, 4.85690932137677564870165477816, 5.57481559369866112358843723543, 6.82022401209932798746809827420, 7.928416604299827156551674594626, 8.760654430341862096546182929312, 9.709897118285139194244753733739, 10.48342080217398943306837401134

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