Properties

Label 2-847-1.1-c1-0-23
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  + 2.70·2-s − 2.80·3-s + 5.30·4-s − 0.445·5-s − 7.58·6-s + 7-s + 8.94·8-s + 4.88·9-s − 1.20·10-s − 14.8·12-s + 0.450·13-s + 2.70·14-s + 1.24·15-s + 13.5·16-s + 4.83·17-s + 13.1·18-s − 1.08·19-s − 2.36·20-s − 2.80·21-s + 4.57·23-s − 25.0·24-s − 4.80·25-s + 1.21·26-s − 5.27·27-s + 5.30·28-s − 1.98·29-s + 3.37·30-s + ⋯
L(s)  = 1  + 1.91·2-s − 1.62·3-s + 2.65·4-s − 0.199·5-s − 3.09·6-s + 0.377·7-s + 3.16·8-s + 1.62·9-s − 0.380·10-s − 4.30·12-s + 0.125·13-s + 0.722·14-s + 0.322·15-s + 3.38·16-s + 1.17·17-s + 3.10·18-s − 0.249·19-s − 0.528·20-s − 0.612·21-s + 0.953·23-s − 5.12·24-s − 0.960·25-s + 0.239·26-s − 1.01·27-s + 1.00·28-s − 0.368·29-s + 0.616·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.267440178\)
\(L(\frac12)\) \(\approx\) \(3.267440178\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70T + 2T^{2} \)
3 \( 1 + 2.80T + 3T^{2} \)
5 \( 1 + 0.445T + 5T^{2} \)
13 \( 1 - 0.450T + 13T^{2} \)
17 \( 1 - 4.83T + 17T^{2} \)
19 \( 1 + 1.08T + 19T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 + 1.98T + 29T^{2} \)
31 \( 1 - 8.25T + 31T^{2} \)
37 \( 1 - 7.31T + 37T^{2} \)
41 \( 1 - 1.77T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 + 3.57T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + 5.92T + 71T^{2} \)
73 \( 1 + 1.65T + 73T^{2} \)
79 \( 1 + 3.60T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 + 5.30T + 97T^{2} \)
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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

−10.70516858954802495578277211285, −9.927612587502116693736709174623, −7.973158871051348606640109074396, −7.11409318054349926520440683803, −6.24174414972478639877358171126, −5.69155622063764222372690020671, −4.88559369033321647331504202175, −4.25183529309960969725066321237, −3.03893462519440549247376285029, −1.39703484169085155131371781379, 1.39703484169085155131371781379, 3.03893462519440549247376285029, 4.25183529309960969725066321237, 4.88559369033321647331504202175, 5.69155622063764222372690020671, 6.24174414972478639877358171126, 7.11409318054349926520440683803, 7.973158871051348606640109074396, 9.927612587502116693736709174623, 10.70516858954802495578277211285

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