Properties

Label 1-837-837.475-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.773 + 0.634i$
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  + (−0.882 + 0.469i)2-s + (0.559 − 0.829i)4-s + (−0.939 + 0.342i)5-s + (−0.719 − 0.694i)7-s + (−0.104 + 0.994i)8-s + (0.669 − 0.743i)10-s + (−0.719 − 0.694i)11-s + (0.848 − 0.529i)13-s + (0.961 + 0.275i)14-s + (−0.374 − 0.927i)16-s + (−0.809 − 0.587i)17-s + (−0.978 − 0.207i)19-s + (−0.241 + 0.970i)20-s + (0.961 + 0.275i)22-s + (0.961 + 0.275i)23-s + ⋯
L(s)  = 1  + (−0.882 + 0.469i)2-s + (0.559 − 0.829i)4-s + (−0.939 + 0.342i)5-s + (−0.719 − 0.694i)7-s + (−0.104 + 0.994i)8-s + (0.669 − 0.743i)10-s + (−0.719 − 0.694i)11-s + (0.848 − 0.529i)13-s + (0.961 + 0.275i)14-s + (−0.374 − 0.927i)16-s + (−0.809 − 0.587i)17-s + (−0.978 − 0.207i)19-s + (−0.241 + 0.970i)20-s + (0.961 + 0.275i)22-s + (0.961 + 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.773 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04367920371 + 0.1220777396i\)
\(L(\frac12)\) \(\approx\) \(0.04367920371 + 0.1220777396i\)
\(L(1)\) \(\approx\) \(0.4531199723 + 0.03475316440i\)
\(L(1)\) \(\approx\) \(0.4531199723 + 0.03475316440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.882 + 0.469i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.719 - 0.694i)T \)
11 \( 1 + (-0.719 - 0.694i)T \)
13 \( 1 + (0.848 - 0.529i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.961 + 0.275i)T \)
29 \( 1 + (-0.882 + 0.469i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.374 + 0.927i)T \)
43 \( 1 + (0.0348 - 0.999i)T \)
47 \( 1 + (-0.615 + 0.788i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (-0.882 - 0.469i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.997 + 0.0697i)T \)
83 \( 1 + (0.0348 - 0.999i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (-0.241 + 0.970i)T \)
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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

−21.52767282009801964542759123544, −20.93240096961719169453943406131, −20.058537359869148285489836175522, −19.38067256728140886912285197793, −18.712779289233398138220796841322, −18.12725612755609952959561531181, −16.90895755740917554758340687178, −16.39290684581780249499463484300, −15.3262945625900608835526297324, −15.20413444208644670239033162893, −13.14789979906592173614709638963, −12.831321247964535650425481585117, −11.93241984006707055155944076681, −11.11972642846200479502671438880, −10.399139847692202799433178540406, −9.22315482846707960877574511115, −8.73252243295697413889172033987, −7.89362079998358659432660720063, −6.94066836466804542094881895913, −6.08148477843636508292191713048, −4.554366167175288912090295939118, −3.72314088295909114991422375024, −2.67388671120843927728330578617, −1.69406718310819130128265897876, −0.09792878075993227829261658527, 0.94979770581252169639501478308, 2.639449514894258539185485028103, 3.492581461814063222457227994068, 4.714824315195339295627060700293, 5.946836721373150429436923517, 6.7765356137805414578482677741, 7.47947040583925756776073397657, 8.32960771805191584169324404648, 9.05854208129700119916808706361, 10.20267038408804923290123637388, 11.000794779694431959022807462246, 11.290692237079727566640287652869, 12.86021297046886428920696030378, 13.51925175315743970448824659313, 14.713159386036579856384933733157, 15.48193661407480364995845411520, 16.05908430872617097772758688382, 16.70710207310766922431400591771, 17.67974541868354715116879250639, 18.68997814960706030460247006274, 18.97154608465467791518267286873, 20.052142678363743490738427596343, 20.36411458072381063592254424960, 21.62444636634635140665372998081, 22.896910798552322606532906096627

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