Properties

Label 1-837-837.14-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.786 - 0.617i$
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  + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (0.913 − 0.406i)10-s + (0.615 + 0.788i)11-s + (−0.719 − 0.694i)13-s + (0.374 + 0.927i)14-s + (0.438 + 0.898i)16-s + (−0.309 + 0.951i)17-s + (−0.104 + 0.994i)19-s + (−0.990 + 0.139i)20-s + (−0.374 − 0.927i)22-s + (0.374 + 0.927i)23-s + ⋯
L(s)  = 1  + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (0.913 − 0.406i)10-s + (0.615 + 0.788i)11-s + (−0.719 − 0.694i)13-s + (0.374 + 0.927i)14-s + (0.438 + 0.898i)16-s + (−0.309 + 0.951i)17-s + (−0.104 + 0.994i)19-s + (−0.990 + 0.139i)20-s + (−0.374 − 0.927i)22-s + (0.374 + 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.786 - 0.617i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.786 - 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002453773170 + 0.007096569792i\)
\(L(\frac12)\) \(\approx\) \(0.002453773170 + 0.007096569792i\)
\(L(1)\) \(\approx\) \(0.5042765478 + 0.03651333916i\)
\(L(1)\) \(\approx\) \(0.5042765478 + 0.03651333916i\)

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 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
11 \( 1 + (0.615 + 0.788i)T \)
13 \( 1 + (-0.719 - 0.694i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.374 + 0.927i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.438 + 0.898i)T \)
43 \( 1 + (-0.241 - 0.970i)T \)
47 \( 1 + (0.997 + 0.0697i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (-0.961 + 0.275i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.882 - 0.469i)T \)
83 \( 1 + (0.241 + 0.970i)T \)
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

−22.197687304096057797538058218579, −21.40728159050910014047275970450, −20.25945771187691938706718085345, −19.784641792681944181897781179382, −18.917106359325941994803948984400, −18.56837502625010752379931483551, −17.27929050301108105113715777111, −16.570742447909123477398086759629, −16.06919498712888358534180863606, −15.2607320456090836589389733263, −14.495451985735465133768464200087, −13.24850480016743345455692930359, −12.163568231357295630538039306725, −11.61798643276650873106879253392, −10.85105391641757193761803997602, −9.33402301402689326774914295463, −9.198679395630421588298283139094, −8.354535043092570540581399007391, −7.2509819205490986853977854084, −6.591759438370374416025988574376, −5.5165400589605155294441142808, −4.54204233007443559900065796189, −3.1535916616794676084243155611, −2.210188418325968610201174464327, −0.772453030623873901997581089438, 0.00331340970288218055616437394, 1.266322259319847947281909637453, 2.48251444450085109126493620323, 3.57610459098841698008537545446, 4.118969985401880684405616169821, 5.918016218015437373342650637744, 6.935072211459199845812964850704, 7.44497659792142577338014445787, 8.205446154593716769619391184782, 9.43359313720337691999644276936, 10.14119867475936891988157012667, 10.739956543928749652058511738207, 11.71580014493243332865700190688, 12.43041694310854942165146903008, 13.28510378782641676490305351540, 14.807825983216520278702472463313, 15.11781086134947907085292982566, 16.18594767003831943358249123333, 17.024135678894903416471680416263, 17.52814413198975468438853286368, 18.61188165127519192574046989113, 19.27059405702027823720610739922, 19.991105537936148524447690667674, 20.31092659977086906988800523010, 21.63894575783983507337772156356

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