Properties

Label 1-837-837.239-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.414 + 0.909i$
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  + (0.241 + 0.970i)2-s + (−0.882 + 0.469i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.809 − 0.587i)10-s + (0.990 − 0.139i)11-s + (0.719 + 0.694i)13-s + (0.615 − 0.788i)14-s + (0.559 − 0.829i)16-s + (0.669 + 0.743i)17-s + (−0.809 − 0.587i)19-s + (0.374 − 0.927i)20-s + (0.374 + 0.927i)22-s + (0.990 + 0.139i)23-s + ⋯
L(s)  = 1  + (0.241 + 0.970i)2-s + (−0.882 + 0.469i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.809 − 0.587i)10-s + (0.990 − 0.139i)11-s + (0.719 + 0.694i)13-s + (0.615 − 0.788i)14-s + (0.559 − 0.829i)16-s + (0.669 + 0.743i)17-s + (−0.809 − 0.587i)19-s + (0.374 − 0.927i)20-s + (0.374 + 0.927i)22-s + (0.990 + 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.414 + 0.909i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.414 + 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6396595961 + 0.9947035868i\)
\(L(\frac12)\) \(\approx\) \(0.6396595961 + 0.9947035868i\)
\(L(1)\) \(\approx\) \(0.7994997841 + 0.5516428816i\)
\(L(1)\) \(\approx\) \(0.7994997841 + 0.5516428816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.241 + 0.970i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
11 \( 1 + (0.990 - 0.139i)T \)
13 \( 1 + (0.719 + 0.694i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.990 + 0.139i)T \)
29 \( 1 + (-0.241 - 0.970i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.438 + 0.898i)T \)
43 \( 1 + (0.241 + 0.970i)T \)
47 \( 1 + (-0.438 - 0.898i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (0.719 + 0.694i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (0.961 + 0.275i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (-0.615 - 0.788i)T \)
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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

−21.939019690820775590073877307105, −20.81335107482204321040444578000, −20.469744933452489197831621667098, −19.43760373759213807792499949063, −18.99867562731838050672459311059, −18.22308570563192929764739328156, −17.08343698121948719293002708618, −16.28486219895400945382384653133, −15.270301759022139064408273291170, −14.65766903747274676156228229802, −13.50012048121335421322628574181, −12.64474975783287403260725876609, −12.2118446199845841259212670965, −11.43231464013951067263657387554, −10.531249435109079135107643471191, −9.416420383587244023202894864696, −8.87771619645973870282735157659, −8.07645364090023396055828713635, −6.65352498033642461376236890030, −5.58884065478426720465368867401, −4.80293503659624626711315700567, −3.65544923916482623554632081626, −3.16983008989668585900038870382, −1.76723550227004559832567284136, −0.67875652633701788966198781831, 0.980133138108032699768323071179, 2.96517094218461255151625617390, 4.012223064992781021278663536081, 4.1962086229382493954364478432, 5.86127133663072666599051024643, 6.64643786012394468699649088373, 7.12207545217246414134861440897, 8.139409278035613246480270427496, 8.97122862311625709634975297447, 9.93474943513447317899347996223, 11.00490475954906941765959977375, 11.82603966127624779056647012287, 12.93143867896097196926031960768, 13.58450343370671083482533813529, 14.5866020471781653505815924939, 14.99717211810912977323902113448, 16.04577836504257852827021795986, 16.679312643779440263384992036607, 17.29171711223580121941715520176, 18.40119856196260536902047956808, 19.25849384355562591079702350508, 19.62093999779568876428825514829, 21.06308803381921131403549135123, 21.8061481653931202799173074311, 22.71065924243936184883431736513

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