Properties

Label 1-837-837.596-r1-0-0
Degree $1$
Conductor $837$
Sign $0.966 + 0.255i$
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 + (0.939 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.966 + 0.255i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (596, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.966 + 0.255i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.892922591 + 0.2457459110i\)
\(L(\frac12)\) \(\approx\) \(1.892922591 + 0.2457459110i\)
\(L(1)\) \(\approx\) \(0.9815908230 + 0.08207468142i\)
\(L(1)\) \(\approx\) \(0.9815908230 + 0.08207468142i\)

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

−21.63866884304507244768718447882, −21.096783443773051291841478485431, −20.39346121939881983084863628801, −19.22365480518371616394965587091, −18.80623822886712549412320745889, −17.758241625707081209995290504229, −17.33964214176151462039310095518, −16.63970373494045662071927261169, −15.608717552192234375333730184292, −14.60876268274444918139344797605, −13.84767771973218980751416488688, −13.004031132951055563295696115884, −11.60508210986416720170291828192, −11.240259240517364400654469391173, −10.47939936835094651827031250729, −9.204151947623397569001665936527, −9.06922191843342633729301327943, −7.82971737795231335670595179866, −6.93065936025306008584784140410, −6.18916047246824713412308596473, −5.02455865320646658557799399982, −3.76621341875640950727023191386, −2.4922557123734752419264605585, −1.836934261241360054564989481345, −0.76422825472008444653200079766, 0.80548733243722593030222169367, 1.88907291687244666202951942864, 2.38017540142963195857979558005, 4.247592094660093314397527431670, 5.31428249534125066690633762247, 6.027058478107856990337225750046, 7.04402945482278270444174755109, 7.975509628151390291950266020335, 8.773001165729384453367971549, 9.44714208231303115671572986901, 10.55328959995739452439501745128, 10.835493860589895804264770210280, 12.27667527609780097409016891998, 12.80795452541450193224799751162, 14.257717306084818927377764784242, 14.78293633828637482449105798258, 15.508001309733753039875539167202, 16.73738193378667681809798921050, 17.36044084213619802787270909414, 17.74667911794632070369582085306, 18.53982058926892477631676823402, 19.56500151361206455753490232460, 20.52871563678180450724404080929, 20.780252415549278850806745384184, 21.82458092292522441223424414379

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