Properties

Label 1-837-837.346-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.179 + 0.983i$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)7-s + (−0.5 + 0.866i)8-s + 10-s + (−0.939 + 0.342i)11-s + (−0.939 − 0.342i)13-s + (0.766 − 0.642i)14-s + (0.766 + 0.642i)16-s + (−0.5 + 0.866i)17-s + 19-s + (0.173 − 0.984i)20-s + (0.173 + 0.984i)22-s + (−0.939 − 0.342i)23-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)7-s + (−0.5 + 0.866i)8-s + 10-s + (−0.939 + 0.342i)11-s + (−0.939 − 0.342i)13-s + (0.766 − 0.642i)14-s + (0.766 + 0.642i)16-s + (−0.5 + 0.866i)17-s + 19-s + (0.173 − 0.984i)20-s + (0.173 + 0.984i)22-s + (−0.939 − 0.342i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.179 + 0.983i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.179 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3878692187 + 0.4651270134i\)
\(L(\frac12)\) \(\approx\) \(0.3878692187 + 0.4651270134i\)
\(L(1)\) \(\approx\) \(0.8470964137 - 0.1006696216i\)
\(L(1)\) \(\approx\) \(0.8470964137 - 0.1006696216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−21.88840828340168913600038276020, −21.27529350514438445685516478650, −20.388812177179915496198329905851, −19.65812829988012393087643121974, −18.248326565347456463937431270454, −17.8568864272126987636047638353, −16.97631804662838480657963676294, −16.231874819251679777918783212101, −15.751854235293842292783060940400, −14.540964643452302533220922275159, −13.87650351892661539393532160822, −13.27236953852177883195194775182, −12.32725737792276993422760303686, −11.46006737209963188254024188877, −10.120392922179464249213838597585, −9.383199861201365516774081433683, −8.42944247861329395444524187382, −7.7116004303169802190212632319, −7.044145851387608299848473772682, −5.70543294186402541458397521921, −4.95363833382299700566509560793, −4.51730194759054220115516137781, −3.191138153341943784729903923355, −1.64174410633210522442847540849, −0.24507094641934762256896426399, 1.771372631372231474909579055702, 2.42737831899390124944084368334, 3.242294634433324313110217160365, 4.49895850023614337431510995060, 5.306299329221433653185040732511, 6.18679608400571433596528088920, 7.60600219545244567876271770475, 8.25905892815020465553864856988, 9.487893498029423749578687860341, 10.253514598293915104794630895662, 10.819770237693268150301321449777, 11.83078427635215557621144129883, 12.33617147308053190295993429985, 13.498735070043635380799211715606, 14.13133681412125689784700853525, 15.09895643355262080494808749486, 15.44380835367887026484811482145, 17.221121800724365179316587404199, 17.851472578966093796433323178111, 18.428740966230678877286826029045, 19.12157887418621186891379553029, 20.0714237734051162907417908858, 20.8084939441733115671319099128, 21.678224030032016833144862298078, 22.15255913703237884058083798315

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