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Properties

Label 1-837-837.445-r1-0-0
Degree $1$
Conductor $837$
Sign $0.441 - 0.897i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.559 − 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.939 − 0.342i)5-s + (−0.882 − 0.469i)7-s + (−0.978 − 0.207i)8-s + (−0.809 + 0.587i)10-s + (−0.848 + 0.529i)11-s + (0.997 − 0.0697i)13-s + (−0.882 + 0.469i)14-s + (−0.719 + 0.694i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (0.0348 + 0.999i)20-s + (−0.0348 + 0.999i)22-s + (−0.848 − 0.529i)23-s + ⋯
L(s)  = 1  + (0.559 − 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.939 − 0.342i)5-s + (−0.882 − 0.469i)7-s + (−0.978 − 0.207i)8-s + (−0.809 + 0.587i)10-s + (−0.848 + 0.529i)11-s + (0.997 − 0.0697i)13-s + (−0.882 + 0.469i)14-s + (−0.719 + 0.694i)16-s + (0.978 + 0.207i)17-s + (−0.809 + 0.587i)19-s + (0.0348 + 0.999i)20-s + (−0.0348 + 0.999i)22-s + (−0.848 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.441 - 0.897i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.441 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9768938533 - 0.6081989263i\)
\(L(\frac12)\) \(\approx\) \(0.9768938533 - 0.6081989263i\)
\(L(1)\) \(\approx\) \(0.7677034954 - 0.4765823802i\)
\(L(1)\) \(\approx\) \(0.7677034954 - 0.4765823802i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.559 - 0.829i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.882 - 0.469i)T \)
11 \( 1 + (-0.848 + 0.529i)T \)
13 \( 1 + (0.997 - 0.0697i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.848 - 0.529i)T \)
29 \( 1 + (-0.559 + 0.829i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.241 + 0.970i)T \)
43 \( 1 + (-0.559 + 0.829i)T \)
47 \( 1 + (-0.241 - 0.970i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (-0.997 + 0.0697i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (0.374 - 0.927i)T \)
83 \( 1 + (-0.438 - 0.898i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−22.320684856143894004625841381330, −21.46134977446798955155002588903, −20.72679778405328719138237318509, −19.56621676334358862055156407292, −18.665596797120702805716048803290, −18.3055731904474992238560778252, −16.9240128794949400342728580470, −16.241086235454575167657757523742, −15.513280773654927893020083905478, −15.20711084990484398809908169535, −13.93738832653296582223798244038, −13.304642339093697592241737100114, −12.42514860898140471421971855761, −11.696121668125441728327274708326, −10.739328191398057330368524541917, −9.53869239657612358666642738223, −8.45792931240191870285457072394, −7.93669827984132517678896781048, −6.93071731215199303846129577978, −6.10433186880131835019477029764, −5.37018839948458202259307993215, −4.07950941291610849677401348181, −3.417522192855261039940057050918, −2.58612491190270334396864322876, −0.39213591185660581454220680522, 0.546630093697602486691982523766, 1.72080163047023703039758809456, 3.09202734263894502532513785065, 3.74760230339265967096806202575, 4.51181161399389910755119725504, 5.61740135153023591715190431761, 6.51734824556085777118193097918, 7.71719727263222749936606554037, 8.57754463083239017470679608734, 9.711510087724746112346552529070, 10.45041551071449512609362499206, 11.13166653355413025951427354247, 12.26866357429765124285994961831, 12.70865119674108134539284205254, 13.39619812764056914531432769244, 14.48209696412216456012837975127, 15.26797863238097855915575852991, 16.11334036161013213158345692356, 16.74525008663363466603395244864, 18.35625199528076703692806265351, 18.62041396852303816298323240860, 19.74889293786378916609188925347, 20.11810315356035622406289475207, 20.93005770340235389648657548738, 21.67545129541268850488830846619

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