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Properties

Label 1-837-837.446-r0-0-0
Degree $1$
Conductor $837$
Sign $0.760 - 0.649i$
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  + (0.615 + 0.788i)2-s + (−0.241 + 0.970i)4-s + (−0.766 + 0.642i)5-s + (0.559 − 0.829i)7-s + (−0.913 + 0.406i)8-s + (−0.978 − 0.207i)10-s + (0.559 − 0.829i)11-s + (−0.990 − 0.139i)13-s + (0.997 − 0.0697i)14-s + (−0.882 − 0.469i)16-s + (−0.809 − 0.587i)17-s + (0.669 − 0.743i)19-s + (−0.438 − 0.898i)20-s + (0.997 − 0.0697i)22-s + (−0.997 + 0.0697i)23-s + ⋯
L(s)  = 1  + (0.615 + 0.788i)2-s + (−0.241 + 0.970i)4-s + (−0.766 + 0.642i)5-s + (0.559 − 0.829i)7-s + (−0.913 + 0.406i)8-s + (−0.978 − 0.207i)10-s + (0.559 − 0.829i)11-s + (−0.990 − 0.139i)13-s + (0.997 − 0.0697i)14-s + (−0.882 − 0.469i)16-s + (−0.809 − 0.587i)17-s + (0.669 − 0.743i)19-s + (−0.438 − 0.898i)20-s + (0.997 − 0.0697i)22-s + (−0.997 + 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (446, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.760 - 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9224198403 - 0.3406001747i\)
\(L(\frac12)\) \(\approx\) \(0.9224198403 - 0.3406001747i\)
\(L(1)\) \(\approx\) \(1.011187493 + 0.2933008705i\)
\(L(1)\) \(\approx\) \(1.011187493 + 0.2933008705i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.615 + 0.788i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.559 - 0.829i)T \)
11 \( 1 + (0.559 - 0.829i)T \)
13 \( 1 + (-0.990 - 0.139i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.997 + 0.0697i)T \)
29 \( 1 + (-0.615 - 0.788i)T \)
37 \( 1 - T \)
41 \( 1 + (0.882 - 0.469i)T \)
43 \( 1 + (0.374 - 0.927i)T \)
47 \( 1 + (-0.848 + 0.529i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (0.615 - 0.788i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.719 - 0.694i)T \)
83 \( 1 + (-0.374 + 0.927i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.438 + 0.898i)T \)
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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

−22.34357793282789025449554967184, −21.36421346180487568668587092708, −20.664657045719607769242817247066, −19.75962169726337314653873871825, −19.517179399587555085603611723809, −18.3155118632887488465201815455, −17.68472909025539024798292465314, −16.495411463404542999030765616344, −15.53099539658418929106286034952, −14.830313057212476777702142246309, −14.287407155039030915265541378485, −12.97953046173089669592613182680, −12.28112838947853768430621945610, −11.88507366223960128371254244529, −11.069469193779026440919509134865, −9.88784668207980932675622176842, −9.171735892008146074411901550, −8.274656696310510420518259469250, −7.19989036957832197046833535299, −5.96737405797225730981484216237, −4.999907692887872975210491090748, −4.41620703346896721197972883130, −3.49047896695202135872418879548, −2.171980488509361758302903979374, −1.48144909264638781516810792340, 0.35276743426179951377670336357, 2.39287698117495177026419257811, 3.46039465062582349614078430767, 4.203077922467496966735561448427, 5.02006121943717096158748489501, 6.191881727356887301524361790051, 7.14851451365750254212180418502, 7.55984698853656430431511498368, 8.481396405537518655485595481860, 9.54634847151922161146539836158, 10.86532807919579950754637450167, 11.53530356625546012935003006534, 12.21833513825954793259908697603, 13.5135177737681765712610225908, 14.05024438257224283223325983979, 14.70380032268907445461755480040, 15.62974964868704961397418963125, 16.203455346340871455178615567796, 17.271173531400235631389422329395, 17.724332589119805072020939737332, 18.81735049412696770857539691105, 19.787595373279817543838167085930, 20.43649721027941113134761025479, 21.556734740021962054759578586724, 22.397913003879470508051878051119

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