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Properties

Label 1-847-847.503-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.968 - 0.249i$
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  + (0.0285 − 0.999i)2-s + (0.809 − 0.587i)3-s + (−0.998 − 0.0570i)4-s + (−0.516 + 0.856i)5-s + (−0.564 − 0.825i)6-s + (−0.0855 + 0.996i)8-s + (0.309 − 0.951i)9-s + (0.841 + 0.540i)10-s + (−0.841 + 0.540i)12-s + (−0.254 − 0.967i)13-s + (0.0855 + 0.996i)15-s + (0.993 + 0.113i)16-s + (−0.985 + 0.170i)17-s + (−0.941 − 0.336i)18-s + (0.696 + 0.717i)19-s + (0.564 − 0.825i)20-s + ⋯
L(s)  = 1  + (0.0285 − 0.999i)2-s + (0.809 − 0.587i)3-s + (−0.998 − 0.0570i)4-s + (−0.516 + 0.856i)5-s + (−0.564 − 0.825i)6-s + (−0.0855 + 0.996i)8-s + (0.309 − 0.951i)9-s + (0.841 + 0.540i)10-s + (−0.841 + 0.540i)12-s + (−0.254 − 0.967i)13-s + (0.0855 + 0.996i)15-s + (0.993 + 0.113i)16-s + (−0.985 + 0.170i)17-s + (−0.941 − 0.336i)18-s + (0.696 + 0.717i)19-s + (0.564 − 0.825i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.968 - 0.249i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.968 - 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1514616974 - 1.196444498i\)
\(L(\frac12)\) \(\approx\) \(0.1514616974 - 1.196444498i\)
\(L(1)\) \(\approx\) \(0.7868841823 - 0.7030052284i\)
\(L(1)\) \(\approx\) \(0.7868841823 - 0.7030052284i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0285 - 0.999i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.516 + 0.856i)T \)
13 \( 1 + (-0.254 - 0.967i)T \)
17 \( 1 + (-0.985 + 0.170i)T \)
19 \( 1 + (0.696 + 0.717i)T \)
23 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (0.466 - 0.884i)T \)
31 \( 1 + (-0.774 + 0.633i)T \)
37 \( 1 + (-0.362 - 0.931i)T \)
41 \( 1 + (0.610 - 0.791i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.941 + 0.336i)T \)
53 \( 1 + (0.993 - 0.113i)T \)
59 \( 1 + (-0.610 - 0.791i)T \)
61 \( 1 + (-0.0285 - 0.999i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (-0.870 + 0.491i)T \)
79 \( 1 + (0.921 - 0.389i)T \)
83 \( 1 + (-0.736 + 0.676i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.516 - 0.856i)T \)
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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

−22.46317124746701124215571899660, −21.696120467277064719373111591604, −21.01601455022052911325603046509, −19.855720914675309576173870620662, −19.53389580720105020093138187485, −18.43549565069694500971737213978, −17.46749215166602213423288921425, −16.4654621284382925906432251007, −16.12560455746060956299446077008, −15.27084521938364921655322349878, −14.680032374881491710570632470681, −13.53439150907905083040539610617, −13.27911785715747888765169832007, −12.01761910520541658803013425392, −10.98565766113356815246107884826, −9.54645872889329772389315566378, −9.20606333487042006761768416896, −8.46468950349271253105400217493, −7.56327127437638812869295290510, −6.86307486013656489491568595876, −5.417813230948111834821070539615, −4.657418717348610304385414620669, −4.08844233947404049458989488516, −2.97657854272533291040520918182, −1.39320473839313387030603693720, 0.50701482937090335305470118266, 1.94301543250936389576291888448, 2.73769068397196476997396401261, 3.47586017369757028741885051845, 4.3112659837616783809525164208, 5.70106182669546440132463935180, 6.88317860820069785513645373160, 7.785421770911383021514705316578, 8.48567042789495315365849185900, 9.43559469120355355944394424899, 10.394493528206033115433678500355, 11.0430690329707802424354459135, 12.150719379872805064463041724179, 12.638655906725612396933680992587, 13.621688859626499572521287834926, 14.334153356871371829369511858787, 14.977377574946153440240067678510, 15.8759124135932690573277441319, 17.48422772330704470741310606631, 18.03488633131538062274111096380, 18.757152847938757794501001650, 19.43863394809112516675193026877, 20.04331610578176143811932488202, 20.70886758164470725103250577103, 21.6477911544510240963781453679

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