Properties

Label 1-837-837.686-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.580 - 0.814i$
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  + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.766 − 0.642i)5-s + (−0.615 + 0.788i)7-s + (−0.669 + 0.743i)8-s + (−0.104 − 0.994i)10-s + (0.374 + 0.927i)11-s + (−0.719 + 0.694i)13-s + (−0.990 + 0.139i)14-s + (−0.997 + 0.0697i)16-s + (0.978 + 0.207i)17-s + (0.913 + 0.406i)19-s + (0.615 − 0.788i)20-s + (−0.374 + 0.927i)22-s + (0.615 + 0.788i)23-s + ⋯
L(s)  = 1  + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.766 − 0.642i)5-s + (−0.615 + 0.788i)7-s + (−0.669 + 0.743i)8-s + (−0.104 − 0.994i)10-s + (0.374 + 0.927i)11-s + (−0.719 + 0.694i)13-s + (−0.990 + 0.139i)14-s + (−0.997 + 0.0697i)16-s + (0.978 + 0.207i)17-s + (0.913 + 0.406i)19-s + (0.615 − 0.788i)20-s + (−0.374 + 0.927i)22-s + (0.615 + 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.580 - 0.814i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (686, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.580 - 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6200126913 + 1.203710589i\)
\(L(\frac12)\) \(\approx\) \(-0.6200126913 + 1.203710589i\)
\(L(1)\) \(\approx\) \(0.8587881676 + 0.7713707986i\)
\(L(1)\) \(\approx\) \(0.8587881676 + 0.7713707986i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.719 + 0.694i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (-0.615 + 0.788i)T \)
11 \( 1 + (0.374 + 0.927i)T \)
13 \( 1 + (-0.719 + 0.694i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.615 + 0.788i)T \)
29 \( 1 + (0.719 + 0.694i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.438 - 0.898i)T \)
43 \( 1 + (-0.241 + 0.970i)T \)
47 \( 1 + (-0.559 - 0.829i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.241 + 0.970i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.882 + 0.469i)T \)
83 \( 1 + (0.719 + 0.694i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
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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

−21.58725485684202612610919365027, −20.47758906088441784890824316682, −19.89785758126465497753825010536, −19.15288987542271909475539193957, −18.704259611388641003393423139630, −17.48664415548152516474330186845, −16.35181678257661069398504797523, −15.71281395791997566291624430570, −14.69752852842294519620132181486, −14.11934456984596746561253025351, −13.31884195785084380740464881075, −12.34038735501657491561931850973, −11.69502276067324517420187373618, −10.775191744324575531477337866945, −10.22158049759621062573064087871, −9.29898467075925149990186283250, −7.93405307810891697342879169254, −7.04067975070030210698987617497, −6.23185379309141769375049237262, −5.14309255339159055112620081595, −4.12307215184045046438185398138, −3.19648975746203665760055474965, −2.84629585018037757449034995001, −1.003764633741140177787716516800, −0.26717022380716008430914953740, 1.5797413519489663905905814653, 3.012577271632872832219989549884, 3.754671708715761636524836057709, 4.883888053661052418149856358, 5.36578079101899774361571794164, 6.63347769164464867211557612978, 7.315597615698995694770344305507, 8.201797273796572851098642838812, 9.129461600992026595550895132669, 9.843780154921176180984035104160, 11.58007404156474710957005030870, 12.14074113054657133845534640692, 12.531412690245167238692852432815, 13.568555392033287060367302291919, 14.61737491988260492990626655343, 15.18838841829383363579835380126, 15.99855120921200402912509753365, 16.61041427587690570972358136356, 17.353932665484002909269444714859, 18.447591807480060491250444499471, 19.3847482701917880039331365915, 20.12282334783248948306232625828, 21.07746532079914617113174864856, 21.76356956116139349078236008099, 22.71338637075098501625312550464

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