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Properties

Label 1-837-837.509-r0-0-0
Degree $1$
Conductor $837$
Sign $0.975 - 0.221i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.766 + 0.642i)5-s + (0.961 + 0.275i)7-s + (0.104 + 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.719 + 0.694i)11-s + (0.882 − 0.469i)13-s + (−0.961 + 0.275i)14-s + (−0.615 − 0.788i)16-s + (−0.104 − 0.994i)17-s + (0.309 − 0.951i)19-s + (0.241 + 0.970i)20-s + (0.241 − 0.970i)22-s + (−0.719 − 0.694i)23-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.766 + 0.642i)5-s + (0.961 + 0.275i)7-s + (0.104 + 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.719 + 0.694i)11-s + (0.882 − 0.469i)13-s + (−0.961 + 0.275i)14-s + (−0.615 − 0.788i)16-s + (−0.104 − 0.994i)17-s + (0.309 − 0.951i)19-s + (0.241 + 0.970i)20-s + (0.241 − 0.970i)22-s + (−0.719 − 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.975 - 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7863720711 - 0.08820830014i\)
\(L(\frac12)\) \(\approx\) \(0.7863720711 - 0.08820830014i\)
\(L(1)\) \(\approx\) \(0.6855993707 + 0.1163097581i\)
\(L(1)\) \(\approx\) \(0.6855993707 + 0.1163097581i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 + (-0.719 + 0.694i)T \)
13 \( 1 + (0.882 - 0.469i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.719 - 0.694i)T \)
29 \( 1 + (0.848 - 0.529i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.990 - 0.139i)T \)
43 \( 1 + (-0.848 + 0.529i)T \)
47 \( 1 + (-0.990 + 0.139i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (0.882 - 0.469i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.438 - 0.898i)T \)
83 \( 1 + (0.0348 + 0.999i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.961 + 0.275i)T \)
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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

−21.63041874944649309424304815406, −21.31857198210005320984682883302, −20.35145796834261177399716032759, −19.95894678154652620141596035726, −18.82361087716702819716715115847, −18.41611662380788374735576014829, −17.39179164922180716801782290770, −16.63142145350261504311787338882, −15.98982368180556735778625122278, −15.19626848091721395174495955633, −13.89825151941810403487462409743, −13.09524254994082743976998040846, −12.08804215714227303574059261511, −11.46960029949989963082027087128, −10.770971563917869510094484706894, −9.92540962442903218285398305323, −8.56026818076003456787022718487, −8.319563697316784394976394569966, −7.62950264326690824308621318415, −6.373841674621506657832227580708, −5.12013943821321705287895304263, −3.99949366897332480794720142043, −3.364608666434146163117359499700, −1.79875668023087529234033357607, −1.09449875633754230811722436798, 0.56457728082755578182290633615, 2.03956095655355139909748517767, 2.93121302014149527519341700529, 4.49844906952304854450080987289, 5.24655913091742574493044525305, 6.43748637305933266345481009310, 7.256651030482703402590509023081, 8.03140040884107681158026244257, 8.552204002058169895350631852514, 9.777283408543379893564166478669, 10.602403241489562524698561567753, 11.33098974417725910811910358975, 11.94204159325272796216245735283, 13.38722079392914232367870453554, 14.373397129692159430611401807725, 15.06927765302778893262339237069, 15.73468066348378687041039736372, 16.27618164630003255568183146667, 17.78439271950360071941303408742, 17.977350824219971041359792849489, 18.61176517294238962284403153470, 19.66332436037683844311582492847, 20.354956210813481797979460749811, 21.03581084871023953944621669022, 22.33150021324901415621168246852

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