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Properties

Label 1-837-837.49-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.203 - 0.979i$
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  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + (0.173 + 0.984i)5-s + (−0.241 − 0.970i)7-s + (−0.104 − 0.994i)8-s + (0.309 − 0.951i)10-s + (0.961 + 0.275i)11-s + (0.0348 − 0.999i)13-s + (−0.241 + 0.970i)14-s + (−0.374 + 0.927i)16-s + (−0.104 − 0.994i)17-s + (0.309 − 0.951i)19-s + (−0.719 + 0.694i)20-s + (−0.719 − 0.694i)22-s + (0.961 − 0.275i)23-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + (0.173 + 0.984i)5-s + (−0.241 − 0.970i)7-s + (−0.104 − 0.994i)8-s + (0.309 − 0.951i)10-s + (0.961 + 0.275i)11-s + (0.0348 − 0.999i)13-s + (−0.241 + 0.970i)14-s + (−0.374 + 0.927i)16-s + (−0.104 − 0.994i)17-s + (0.309 − 0.951i)19-s + (−0.719 + 0.694i)20-s + (−0.719 − 0.694i)22-s + (0.961 − 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.203 - 0.979i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.203 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5045135817 - 0.6199048416i\)
\(L(\frac12)\) \(\approx\) \(0.5045135817 - 0.6199048416i\)
\(L(1)\) \(\approx\) \(0.6885691500 - 0.2190232600i\)
\(L(1)\) \(\approx\) \(0.6885691500 - 0.2190232600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.882 - 0.469i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.241 - 0.970i)T \)
11 \( 1 + (0.961 + 0.275i)T \)
13 \( 1 + (0.0348 - 0.999i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.961 - 0.275i)T \)
29 \( 1 + (-0.882 - 0.469i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.615 + 0.788i)T \)
43 \( 1 + (-0.882 - 0.469i)T \)
47 \( 1 + (-0.615 - 0.788i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (0.848 - 0.529i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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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

−22.331897938859873992134733907998, −21.37364699499472795011613980355, −20.75529762343534746740484411895, −19.63590452001330049364085647071, −19.20870708821085579339946225887, −18.39514328253122869601067987846, −17.41736763676477664729681976157, −16.670594772061315536135700868322, −16.293038639737747773047469326883, −15.23012307439075070462749769592, −14.55744594255463808738277335061, −13.55080946405089553872782513355, −12.36700030907487597318065818574, −11.80619471553926227459317797122, −10.80389422390397832965642328816, −9.599126383675128402135426545405, −9.035234750298115306472964815438, −8.59563503877284953968058861192, −7.49153733677273810509241691278, −6.337688785527641613888350761961, −5.79885529652397359649027213028, −4.81098323447981019378608412812, −3.527128023048696106983135688744, −1.918146065467275326131662051360, −1.37201365101670772320385811916, 0.5025389702577698531989461974, 1.73935379998712100750021389798, 3.0198118875514706592916315192, 3.470399632412218046722402662451, 4.80037183509392129141855940171, 6.41038175519214607330489999977, 7.02223653596078898809960151485, 7.6197949128432852050679905330, 8.8334579462671667482738174945, 9.77801810752445626580115826695, 10.26827503621844622239216002312, 11.23573096271944214079305984908, 11.68981942284631905885344247900, 13.05073078518536735266013934215, 13.613375359148786386960911877646, 14.79828174940954553043119063031, 15.50848157205058623651102826940, 16.61572369447915314718513361731, 17.285500866095057659797833973601, 17.928535280699632198216964746669, 18.71555276151544852113382623032, 19.56075241630663658308406314644, 20.17833444643996696792072267401, 20.83032260601456799403580545404, 22.05072770536271735926581986497

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