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Properties

Label 1-837-837.565-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: $\chi_{837} (565, \cdot )$
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

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

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