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Properties

Label 1-847-847.710-r0-0-0
Degree $1$
Conductor $847$
Sign $0.309 + 0.950i$
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  + (0.683 − 0.730i)2-s + (−0.669 + 0.743i)3-s + (−0.0665 − 0.997i)4-s + (−0.988 − 0.151i)5-s + (0.0855 + 0.996i)6-s + (−0.774 − 0.633i)8-s + (−0.104 − 0.994i)9-s + (−0.786 + 0.618i)10-s + (0.786 + 0.618i)12-s + (−0.466 − 0.884i)13-s + (0.774 − 0.633i)15-s + (−0.991 + 0.132i)16-s + (−0.948 + 0.318i)17-s + (−0.797 − 0.603i)18-s + (−0.595 + 0.803i)19-s + (−0.0855 + 0.996i)20-s + ⋯
L(s)  = 1  + (0.683 − 0.730i)2-s + (−0.669 + 0.743i)3-s + (−0.0665 − 0.997i)4-s + (−0.988 − 0.151i)5-s + (0.0855 + 0.996i)6-s + (−0.774 − 0.633i)8-s + (−0.104 − 0.994i)9-s + (−0.786 + 0.618i)10-s + (0.786 + 0.618i)12-s + (−0.466 − 0.884i)13-s + (0.774 − 0.633i)15-s + (−0.991 + 0.132i)16-s + (−0.948 + 0.318i)17-s + (−0.797 − 0.603i)18-s + (−0.595 + 0.803i)19-s + (−0.0855 + 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (710, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3685851801 + 0.2676601765i\)
\(L(\frac12)\) \(\approx\) \(0.3685851801 + 0.2676601765i\)
\(L(1)\) \(\approx\) \(0.7598859881 - 0.1882686497i\)
\(L(1)\) \(\approx\) \(0.7598859881 - 0.1882686497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.683 - 0.730i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (-0.988 - 0.151i)T \)
13 \( 1 + (-0.466 - 0.884i)T \)
17 \( 1 + (-0.948 + 0.318i)T \)
19 \( 1 + (-0.595 + 0.803i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (0.736 + 0.676i)T \)
31 \( 1 + (-0.272 + 0.962i)T \)
37 \( 1 + (0.640 - 0.768i)T \)
41 \( 1 + (0.516 - 0.856i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.797 + 0.603i)T \)
53 \( 1 + (-0.991 - 0.132i)T \)
59 \( 1 + (-0.483 + 0.875i)T \)
61 \( 1 + (-0.290 + 0.956i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (0.997 + 0.0760i)T \)
79 \( 1 + (-0.449 + 0.893i)T \)
83 \( 1 + (0.941 + 0.336i)T \)
89 \( 1 + (0.995 + 0.0950i)T \)
97 \( 1 + (0.362 - 0.931i)T \)
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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

−22.06428424261284865116754556166, −21.64845231559474144480152741827, −20.30912680649872463715888501180, −19.44706347902846448443742735058, −18.71156085921538959979083591972, −17.76657838360301852535789279326, −17.02394133879776769671792955854, −16.35433539565515396712511441631, −15.47084031256923096693513406410, −14.84112458801527694016078033335, −13.697043924855764108849443896317, −13.13236292393557643894892402244, −12.21642166832555261599559024673, −11.48746615602505492607808064456, −11.069737966225622550273466316620, −9.34909480864008489630662827568, −8.28260855200816597977923361078, −7.561751002279911427109063412528, −6.759427611650057128600760735864, −6.29046064282028130025437697603, −4.80674983052179523995577050548, −4.55012403297182241774704495265, −3.16211629137291948647775047755, −2.09365274314537774410500584248, −0.200436883899908662233682432251, 1.09667373867876729753759024958, 2.73523218889264736434523195187, 3.61373451353045404312082765275, 4.45964608834171033143687061241, 5.03503016357792010194204938219, 6.081555447122447569660303623, 6.99647711125301772308195252933, 8.40963817165028627445310369762, 9.25971448567019506612728860797, 10.50540491568543200156154690733, 10.72873084949340724656278668003, 11.68139030247658005965369385312, 12.49734450433547098728150674556, 12.90363784887245370628285244178, 14.49267119650948965435494466537, 14.92348397926982508425013552742, 15.75719157639376265537857063665, 16.39787524767184757714214002786, 17.540581523477849892070861367315, 18.34393632214268442822984417304, 19.445435905353277195937267326642, 19.97259832580872697690194206344, 20.77976762469030665120951140758, 21.5005545034247340725427100063, 22.36874267629681915003918264606

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