Properties

Label 2-847-1.1-c1-0-1
Degree $2$
Conductor $847$
Sign $1$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·2-s − 1.23·3-s + 3.00·4-s − 2·5-s + 2.76·6-s − 7-s − 2.23·8-s − 1.47·9-s + 4.47·10-s − 3.70·12-s − 3.23·13-s + 2.23·14-s + 2.47·15-s − 0.999·16-s + 3.23·17-s + 3.29·18-s − 6.47·19-s − 6.00·20-s + 1.23·21-s + 2.47·23-s + 2.76·24-s − 25-s + 7.23·26-s + 5.52·27-s − 3.00·28-s − 8.47·29-s − 5.52·30-s + ⋯
L(s)  = 1  − 1.58·2-s − 0.713·3-s + 1.50·4-s − 0.894·5-s + 1.12·6-s − 0.377·7-s − 0.790·8-s − 0.490·9-s + 1.41·10-s − 1.07·12-s − 0.897·13-s + 0.597·14-s + 0.638·15-s − 0.249·16-s + 0.784·17-s + 0.775·18-s − 1.48·19-s − 1.34·20-s + 0.269·21-s + 0.515·23-s + 0.564·24-s − 0.200·25-s + 1.41·26-s + 1.06·27-s − 0.566·28-s − 1.57·29-s − 1.00·30-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.1861895667\)
\(L(\frac12)\) \(\approx\) \(0.1861895667\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 2.23T + 2T^{2} \)
3 \( 1 + 1.23T + 3T^{2} \)
5 \( 1 + 2T + 5T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 - 3.23T + 17T^{2} \)
19 \( 1 + 6.47T + 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + 8.47T + 29T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 2.76T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 - 7.23T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 0.763T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−10.20363652513840787370961731284, −9.294429081734798438084641412023, −8.566587085246302913163866134024, −7.71622972342992786620380554344, −7.08931604260846426594891339846, −6.11779280300154210398300402993, −4.99157625858409874459487405215, −3.64672976393765158572239912627, −2.19101814798841411671126659391, −0.43099911482503930745366432678, 0.43099911482503930745366432678, 2.19101814798841411671126659391, 3.64672976393765158572239912627, 4.99157625858409874459487405215, 6.11779280300154210398300402993, 7.08931604260846426594891339846, 7.71622972342992786620380554344, 8.566587085246302913163866134024, 9.294429081734798438084641412023, 10.20363652513840787370961731284

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