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Properties

Label 1-847-847.61-r0-0-0
Degree $1$
Conductor $847$
Sign $0.634 + 0.772i$
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  + (−0.548 − 0.836i)2-s + (0.978 + 0.207i)3-s + (−0.398 + 0.917i)4-s + (−0.999 + 0.0380i)5-s + (−0.362 − 0.931i)6-s + (0.985 − 0.170i)8-s + (0.913 + 0.406i)9-s + (0.580 + 0.814i)10-s + (−0.580 + 0.814i)12-s + (−0.870 − 0.491i)13-s + (−0.985 − 0.170i)15-s + (−0.683 − 0.730i)16-s + (−0.761 + 0.647i)17-s + (−0.161 − 0.986i)18-s + (−0.851 + 0.524i)19-s + (0.362 − 0.931i)20-s + ⋯
L(s)  = 1  + (−0.548 − 0.836i)2-s + (0.978 + 0.207i)3-s + (−0.398 + 0.917i)4-s + (−0.999 + 0.0380i)5-s + (−0.362 − 0.931i)6-s + (0.985 − 0.170i)8-s + (0.913 + 0.406i)9-s + (0.580 + 0.814i)10-s + (−0.580 + 0.814i)12-s + (−0.870 − 0.491i)13-s + (−0.985 − 0.170i)15-s + (−0.683 − 0.730i)16-s + (−0.761 + 0.647i)17-s + (−0.161 − 0.986i)18-s + (−0.851 + 0.524i)19-s + (0.362 − 0.931i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.634 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8119324130 + 0.3838464196i\)
\(L(\frac12)\) \(\approx\) \(0.8119324130 + 0.3838464196i\)
\(L(1)\) \(\approx\) \(0.8348154477 - 0.07064994363i\)
\(L(1)\) \(\approx\) \(0.8348154477 - 0.07064994363i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.548 - 0.836i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (-0.999 + 0.0380i)T \)
13 \( 1 + (-0.870 - 0.491i)T \)
17 \( 1 + (-0.761 + 0.647i)T \)
19 \( 1 + (-0.851 + 0.524i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (0.948 + 0.318i)T \)
37 \( 1 + (-0.217 + 0.976i)T \)
41 \( 1 + (-0.254 + 0.967i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.161 + 0.986i)T \)
53 \( 1 + (-0.683 + 0.730i)T \)
59 \( 1 + (-0.964 - 0.263i)T \)
61 \( 1 + (0.449 + 0.893i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (-0.999 + 0.0190i)T \)
79 \( 1 + (-0.272 + 0.962i)T \)
83 \( 1 + (0.0855 + 0.996i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
97 \( 1 + (0.466 + 0.884i)T \)
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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

−22.12926478410372500750134201531, −21.00550748988441568191426565405, −20.00576200716839477467502137888, −19.45410153142172476601899419869, −18.98711912943584345423253821374, −18.08071757418023680380724937120, −17.171729602404778321336060724613, −16.20266094024776652661972994697, −15.51427310112487769340965370668, −14.88648764770122664665040696125, −14.224928686994106911539109663751, −13.28011959842260530475321998585, −12.390561037737931039312111779355, −11.24271892597665501377644543462, −10.301218240049836262318975870083, −9.16402972347684473513578237419, −8.77318447888703515311406753136, −7.84934685345276286818587365863, −7.087247537022570338876079146088, −6.60056808436770175386856935457, −4.85157795826107172265139582174, −4.38124365805190147498974698793, −3.037143905758580297742716689745, −1.92704717410300938381713587507, −0.46366222824443314544371054428, 1.25107285273929096876267358413, 2.516682604769231922931409428119, 3.144774757668761813412041608448, 4.19764875391607553171336765576, 4.71431589588182620998182057268, 6.716623758649771409331085984597, 7.668928617141934522961997209206, 8.28427329735521475666349666288, 8.91165056211081728322679263345, 9.99551607088395423407421886320, 10.58751453650730764973948680506, 11.55809820774829577716977493886, 12.49820807196010851904009555013, 13.055919811414715005645585735631, 14.10367191818281716957877949049, 15.13457696670538691557692669974, 15.60298896924372018419599174511, 16.77357620997503716383085065319, 17.47924777372448699418313393968, 18.684498982040037728963248147265, 19.26613487099322225543676614650, 19.69449869065499902626667241446, 20.4780454904229567086809653071, 21.17023901134220986862775893288, 22.00804442766371510975506216137

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