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Properties

Label 1-837-837.40-r0-0-0
Degree $1$
Conductor $837$
Sign $0.255 + 0.966i$
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  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.939 − 0.342i)5-s + (0.990 − 0.139i)7-s + (0.669 + 0.743i)8-s + (−0.809 − 0.587i)10-s + (−0.374 + 0.927i)11-s + (−0.241 + 0.970i)13-s + (0.990 + 0.139i)14-s + (0.438 + 0.898i)16-s + (0.669 + 0.743i)17-s + (−0.809 − 0.587i)19-s + (−0.615 − 0.788i)20-s + (−0.615 + 0.788i)22-s + (−0.374 − 0.927i)23-s + ⋯
L(s)  = 1  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.939 − 0.342i)5-s + (0.990 − 0.139i)7-s + (0.669 + 0.743i)8-s + (−0.809 − 0.587i)10-s + (−0.374 + 0.927i)11-s + (−0.241 + 0.970i)13-s + (0.990 + 0.139i)14-s + (0.438 + 0.898i)16-s + (0.669 + 0.743i)17-s + (−0.809 − 0.587i)19-s + (−0.615 − 0.788i)20-s + (−0.615 + 0.788i)22-s + (−0.374 − 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.255 + 0.966i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.255 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923864618 + 1.481737320i\)
\(L(\frac12)\) \(\approx\) \(1.923864618 + 1.481737320i\)
\(L(1)\) \(\approx\) \(1.654770732 + 0.5554066913i\)
\(L(1)\) \(\approx\) \(1.654770732 + 0.5554066913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.961 + 0.275i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.990 - 0.139i)T \)
11 \( 1 + (-0.374 + 0.927i)T \)
13 \( 1 + (-0.241 + 0.970i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.374 - 0.927i)T \)
29 \( 1 + (0.961 + 0.275i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + (0.961 + 0.275i)T \)
47 \( 1 + (-0.997 - 0.0697i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.241 + 0.970i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (0.848 - 0.529i)T \)
83 \( 1 + (-0.719 + 0.694i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (0.990 - 0.139i)T \)
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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

−21.93383317659545566075686916720, −21.22014464335129715794052976543, −20.56661575350788901116343536670, −19.674547120764334768100274981952, −18.990001743126035776725859062102, −18.19397241147416356510359038080, −17.053690498832600658958352583130, −15.89692629052977604304778609761, −15.54202971252812212663793059389, −14.49309006273739667334945508487, −14.11763822487352982475100246280, −12.96687845989825298580982618398, −12.11360604395162462230410330977, −11.46280015747109027258725163276, −10.794400493791968696503708325690, −10.003814322131315794570928498519, −8.32731587402941902978458085627, −7.842407527668538724562939850119, −6.86106654829397637996410437840, −5.63899692460818734308910952690, −5.070441914249855643471854400048, −3.940426290646904068525328416627, −3.21484769791851898856691765443, −2.23470705839249947750310554384, −0.822729165859833474119877169208, 1.51481355856446167932866228812, 2.50902409226338857496024095036, 3.84183952718687467408215827083, 4.58774326563050504542883016346, 4.9998874432848501579400470852, 6.429725578997054614594337820901, 7.2268601119803842919019969162, 8.04842864417302033440680680482, 8.682771803172879124666924711278, 10.297506042415484678932086305516, 11.07162469390645447912232985306, 12.06215642971139601995664867975, 12.33358224463858038100440293940, 13.42270373030427205339392636228, 14.4366389655422727848260922512, 14.95051846128614863533905731196, 15.64647052500630121424592303013, 16.65366314788239876121437460090, 17.169634999220322028135013233321, 18.27730226706438925097730038729, 19.41745351446851493691509973492, 20.05756632208104054498217865741, 20.969949497254873964248918679799, 21.349487398421559012316637325303, 22.45997051050559831843237753518

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