Properties

Label 2-847-77.48-c0-0-1
Degree $2$
Conductor $847$
Sign $-0.995 + 0.0913i$
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  + (−1.30 − 0.951i)2-s + (0.500 + 1.53i)4-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.499 + 1.53i)14-s + (0.5 + 1.53i)18-s − 1.61·23-s + (0.309 − 0.951i)25-s + (1.30 − 0.951i)28-s + (−0.190 − 0.587i)29-s − 0.999·32-s + (0.499 − 1.53i)36-s + (−0.5 − 1.53i)37-s − 0.618·43-s + ⋯
L(s)  = 1  + (−1.30 − 0.951i)2-s + (0.500 + 1.53i)4-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.499 + 1.53i)14-s + (0.5 + 1.53i)18-s − 1.61·23-s + (0.309 − 0.951i)25-s + (1.30 − 0.951i)28-s + (−0.190 − 0.587i)29-s − 0.999·32-s + (0.499 − 1.53i)36-s + (−0.5 − 1.53i)37-s − 0.618·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.995 + 0.0913i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ -0.995 + 0.0913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3091916357\)
\(L(\frac12)\) \(\approx\) \(0.3091916357\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.999918916586626457518071557147, −9.347189786269145117859156957906, −8.441270364421093327515619238717, −7.81756205265229096339741226985, −6.78344233225989769102366057380, −5.75962347876640331536145448310, −4.13143090436035810498936726619, −3.22508619898816023538158437224, −2.04569229335586695001424957120, −0.45754450192027808659234897582, 1.87488074131748201667848419479, 3.26398044503351980067398821437, 5.07012903673410262345504589415, 5.87556942901334309504114122056, 6.57562388831626833846385822526, 7.64397418199673817283771218072, 8.342472638051968131771842325024, 8.915279016991001173459354428843, 9.712694596399048929058284963843, 10.45590093473702290302599347479

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