Properties

Label 2-847-7.6-c0-0-1
Degree $2$
Conductor $847$
Sign $1$
Analytic cond. $0.422708$
Root an. cond. $0.650160$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 1.61·4-s + 7-s − 8-s + 9-s − 1.61·14-s − 1.61·18-s − 1.61·23-s + 25-s + 1.61·28-s + 0.618·29-s + 32-s + 1.61·36-s − 1.61·37-s + 0.618·43-s + 2.61·46-s + 49-s − 1.61·50-s + 0.618·53-s − 56-s − 1.00·58-s + 63-s − 1.61·64-s + 0.618·67-s + 0.618·71-s − 72-s + 2.61·74-s + ⋯
L(s)  = 1  − 1.61·2-s + 1.61·4-s + 7-s − 8-s + 9-s − 1.61·14-s − 1.61·18-s − 1.61·23-s + 25-s + 1.61·28-s + 0.618·29-s + 32-s + 1.61·36-s − 1.61·37-s + 0.618·43-s + 2.61·46-s + 49-s − 1.61·50-s + 0.618·53-s − 56-s − 1.00·58-s + 63-s − 1.61·64-s + 0.618·67-s + 0.618·71-s − 72-s + 2.61·74-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (727, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5541335636\)
\(L(\frac12)\) \(\approx\) \(0.5541335636\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.36300542543562407087495051759, −9.542526919645445145299868730228, −8.656433858379158665847379669118, −8.029645922169226591581852806740, −7.29147343091261307844789775485, −6.50769438578729732944439167979, −5.11287903436324213387937648395, −4.04389177856401974827877511335, −2.26214367040914603615949696135, −1.28378326549801024809477513366, 1.28378326549801024809477513366, 2.26214367040914603615949696135, 4.04389177856401974827877511335, 5.11287903436324213387937648395, 6.50769438578729732944439167979, 7.29147343091261307844789775485, 8.029645922169226591581852806740, 8.656433858379158665847379669118, 9.542526919645445145299868730228, 10.36300542543562407087495051759

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