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Properties

Label 1-837-837.7-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.964 + 0.262i$
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  + (−0.241 + 0.970i)2-s + (−0.882 − 0.469i)4-s + (0.766 + 0.642i)5-s + (−0.615 + 0.788i)7-s + (0.669 − 0.743i)8-s + (−0.809 + 0.587i)10-s + (0.990 + 0.139i)11-s + (−0.719 + 0.694i)13-s + (−0.615 − 0.788i)14-s + (0.559 + 0.829i)16-s + (0.669 − 0.743i)17-s + (−0.809 + 0.587i)19-s + (−0.374 − 0.927i)20-s + (−0.374 + 0.927i)22-s + (0.990 − 0.139i)23-s + ⋯
L(s)  = 1  + (−0.241 + 0.970i)2-s + (−0.882 − 0.469i)4-s + (0.766 + 0.642i)5-s + (−0.615 + 0.788i)7-s + (0.669 − 0.743i)8-s + (−0.809 + 0.587i)10-s + (0.990 + 0.139i)11-s + (−0.719 + 0.694i)13-s + (−0.615 − 0.788i)14-s + (0.559 + 0.829i)16-s + (0.669 − 0.743i)17-s + (−0.809 + 0.587i)19-s + (−0.374 − 0.927i)20-s + (−0.374 + 0.927i)22-s + (0.990 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.964 + 0.262i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.964 + 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1500833460 + 1.124368711i\)
\(L(\frac12)\) \(\approx\) \(0.1500833460 + 1.124368711i\)
\(L(1)\) \(\approx\) \(0.6796310122 + 0.6459834659i\)
\(L(1)\) \(\approx\) \(0.6796310122 + 0.6459834659i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.241 + 0.970i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.615 + 0.788i)T \)
11 \( 1 + (0.990 + 0.139i)T \)
13 \( 1 + (-0.719 + 0.694i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.990 - 0.139i)T \)
29 \( 1 + (-0.241 + 0.970i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.438 + 0.898i)T \)
43 \( 1 + (-0.241 + 0.970i)T \)
47 \( 1 + (0.438 - 0.898i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.719 + 0.694i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.882 + 0.469i)T \)
83 \( 1 + (0.961 - 0.275i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.615 + 0.788i)T \)
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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

−21.73018759148575291270477810868, −20.83778942951760308753596474172, −20.22691452092878143263945735109, −19.40267195851699864449239026788, −18.96029187566913823813789894342, −17.4670589005852368754816122572, −17.184141853548315875971248347621, −16.70470411020899162731359685022, −15.236454349870506893695542889871, −14.14745710125660356064733486242, −13.48243237032492101138191224642, −12.7073338268536998252447953481, −12.18546836819276951363919991380, −10.93348306399317454575988212727, −10.23745256065872004899835816119, −9.5151528633606509619140959261, −8.84318526319650582627834124987, −7.82129275773717078313225977614, −6.68093896593251401658731830954, −5.58091295578476327154976370312, −4.55412720834421952848326323373, −3.714806186022220247833775021503, −2.67893518283423452603762840974, −1.54173465847448890713569028915, −0.60031600293699409325929158233, 1.44401349566117401588036896541, 2.64122301737102665801529918325, 3.824576209763934047288444558175, 5.07363927554826843025174135737, 5.84033007348694642603554071892, 6.74121793074368976030427631283, 7.12331987267917703984695689580, 8.53988415574876636201849537641, 9.39596431744660514624036945112, 9.72288380470845624602424470120, 10.84198615950249625730057505753, 12.07543492699767861036079427375, 12.90581921252954826049839805227, 13.91160919419969542634873387147, 14.69783259266194593213307119304, 14.96794575258932686143172937276, 16.35569158665962441232927201047, 16.73908779504226724043061066368, 17.65208517547441036491569780991, 18.49587041569798185913621570069, 19.01807344545603454587684198233, 19.76949354892620337007896002771, 21.35831988606228762624325651574, 21.752728360649498641909085774168, 22.81416001956067169518644699284

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