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Properties

Label 2-847-77.20-c0-0-2
Degree $2$
Conductor $847$
Sign $-0.605 + 0.795i$
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  + (0.5 − 1.53i)2-s + (−1.30 − 0.951i)4-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (1.30 − 0.951i)14-s + (−1.30 − 0.951i)18-s − 1.61·23-s + (−0.809 + 0.587i)25-s + (−0.5 − 1.53i)28-s + (0.5 + 0.363i)29-s − 0.999·32-s + (−1.30 + 0.951i)36-s + (1.30 + 0.951i)37-s − 0.618·43-s + ⋯
L(s)  = 1  + (0.5 − 1.53i)2-s + (−1.30 − 0.951i)4-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (1.30 − 0.951i)14-s + (−1.30 − 0.951i)18-s − 1.61·23-s + (−0.809 + 0.587i)25-s + (−0.5 − 1.53i)28-s + (0.5 + 0.363i)29-s − 0.999·32-s + (−1.30 + 0.951i)36-s + (1.30 + 0.951i)37-s − 0.618·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.296540704\)
\(L(\frac12)\) \(\approx\) \(1.296540704\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−10.11759844084529765792478259736, −9.653127319194560612156646817456, −8.689304508533416024102620010488, −7.73073220431261540887073258294, −6.38761227070568781108728874996, −5.37317498533740963048668861984, −4.38550393331974952305843492107, −3.59848117974718957243064951997, −2.43446503655122651047927255317, −1.38915356027729698799990328562, 2.05005894441119168563851954967, 4.03093627975874271178099693766, 4.56979869034292337630592207281, 5.53175816193951444730174524950, 6.32887934822636847160132571995, 7.40011905615400127809694520064, 7.88809071221403575077977537890, 8.456244803529155901168200967212, 9.784536502899081514294031648811, 10.61752287734425655615218812922

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