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Properties

Label 1-936-936.419-r0-0-0
Degree $1$
Conductor $936$
Sign $0.822 - 0.568i$
Analytic cond. $4.34676$
Root an. cond. $4.34676$
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  + (−0.5 − 0.866i)5-s − 7-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + 23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s + (0.5 − 0.866i)37-s − 41-s + 43-s + (−0.5 + 0.866i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s − 7-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + 23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s + (0.5 − 0.866i)37-s − 41-s + 43-s + (−0.5 + 0.866i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(4.34676\)
Root analytic conductor: \(4.34676\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 936,\ (0:\ ),\ 0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.071142756 - 0.3339584313i\)
\(L(\frac12)\) \(\approx\) \(1.071142756 - 0.3339584313i\)
\(L(1)\) \(\approx\) \(0.9100187605 - 0.1166087564i\)
\(L(1)\) \(\approx\) \(0.9100187605 - 0.1166087564i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−22.07983554408924455874067072243, −21.21980317806769961185565938517, −20.21316970517552975103727501577, −19.37504477323069463533876721428, −18.79619097739816269558131264154, −18.32303609275593616911905155095, −16.85167347576685223034674579732, −16.515052733180766368284244914117, −15.50146312723627872076987389103, −14.80004290601196148626071961680, −13.953522968582530027478427879287, −13.1781879290695255735610010530, −12.141939551834944247039876732662, −11.4561055907509446448458380482, −10.58208061657678903068111765977, −9.79942758552331709861771945190, −8.8886977612207375496592509937, −7.89617722689775068558106460311, −6.93895782203411928769285546445, −6.35184915855819981724051012397, −5.37914458128151618628383495996, −3.95275943730339314959250778347, −3.3277477301152965024326929340, −2.52126259451725171746328975855, −0.85760884878864184972787521527, 0.69682349405762766685852268368, 1.94543704207397857206616752788, 3.23736144230504488962765833795, 4.12440681603793684349103787307, 4.918494001714711437644434049395, 6.04045313503950681423648694234, 6.916066218698095660554321651100, 7.783754517150286821281724338898, 8.86235568959701883931486428633, 9.39355392059511675541757004283, 10.32978190938740791894188115289, 11.368215556872080689846391213, 12.3660889011987634891788454912, 12.75804386867975406326104540201, 13.55050897029996200195654929581, 14.824797083782543690153909954279, 15.41650355806455661361515723931, 16.23368705062617402796298402109, 17.03764305594674413657702755997, 17.53686439877749065715474479095, 18.92208931606933355906252822224, 19.46964495737812264457043456259, 20.02507992079837253950082270302, 20.949865459826741108351030870767, 21.6900102809011674212587876716

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