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Properties

Label 1-847-847.10-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.700 + 0.713i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (0.995 + 0.0950i)2-s + (0.5 + 0.866i)3-s + (0.981 + 0.189i)4-s + (−0.235 + 0.971i)5-s + (0.415 + 0.909i)6-s + (0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.327 + 0.945i)10-s + (0.327 + 0.945i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.888 − 0.458i)17-s + (−0.580 + 0.814i)18-s + (−0.888 + 0.458i)19-s + (−0.415 + 0.909i)20-s + ⋯
L(s)  = 1  + (0.995 + 0.0950i)2-s + (0.5 + 0.866i)3-s + (0.981 + 0.189i)4-s + (−0.235 + 0.971i)5-s + (0.415 + 0.909i)6-s + (0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.327 + 0.945i)10-s + (0.327 + 0.945i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.888 − 0.458i)17-s + (−0.580 + 0.814i)18-s + (−0.888 + 0.458i)19-s + (−0.415 + 0.909i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.700 + 0.713i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.700 + 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.107265086 + 2.637451448i\)
\(L(\frac12)\) \(\approx\) \(1.107265086 + 2.637451448i\)
\(L(1)\) \(\approx\) \(1.617521736 + 1.210703811i\)
\(L(1)\) \(\approx\) \(1.617521736 + 1.210703811i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.995 + 0.0950i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.235 + 0.971i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (-0.888 + 0.458i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (0.327 - 0.945i)T \)
37 \( 1 + (0.981 - 0.189i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.580 - 0.814i)T \)
53 \( 1 + (0.928 - 0.371i)T \)
59 \( 1 + (0.995 - 0.0950i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.786 + 0.618i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.79958248963228591423767686917, −20.87425197756615242562694551048, −20.26159705726595261131027071797, −19.59894020780432687295403215497, −19.06412808334422799422448303794, −17.6378746661532186735243394362, −17.06588013250552221750861655525, −15.97298716428952308479020829434, −15.111924589785079677328412342344, −14.55019848368787753995665109417, −13.43090322708786535547234672413, −12.82553055287207331482082452138, −12.51775943468491276251093760585, −11.48773400593146485107474161303, −10.60085255055121403401110736169, −9.22817560878214424106105576217, −8.428170864145358327259706518158, −7.51353677278975791909481198318, −6.73008905487008805751968954728, −5.7358798591819632753415341442, −4.81994674504402566070075465596, −3.920879416260855609320834559788, −2.797269149730135708151348896339, −1.96948311335617405580578902792, −0.81237483158017336173238263596, 2.22716781339788440673737973151, 2.6507699314379055837600469705, 3.88128308627068950001456949979, 4.30935040426622626794310977014, 5.41077269662819400012225617447, 6.45880667921629693733497730094, 7.2791943750309845056860102525, 8.147030156200756926005554358900, 9.40329827138751740106601686621, 10.22115909748058917655599106430, 11.31569566465367177463562094092, 11.43267372075043333105362075248, 12.94068996922606331528706107648, 13.669007250044656374946097156375, 14.67493691003088187000666279824, 14.85119230062501127225573643330, 15.69381838175421529689239890522, 16.54620146532430205367342106874, 17.29955857303363001062416056665, 18.72633111664305811441202883405, 19.47634685519631399094069265557, 20.10364646053021496658477921217, 21.27078932507339956257417577509, 21.44535109906236562810462157777, 22.59633905267534004946201278342

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