Properties

Label 1-837-837.506-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.634 - 0.773i$
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  + (−0.848 − 0.529i)2-s + (0.438 + 0.898i)4-s + (−0.173 + 0.984i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (0.669 − 0.743i)10-s + (0.241 − 0.970i)11-s + (0.0348 + 0.999i)13-s + (0.719 − 0.694i)14-s + (−0.615 + 0.788i)16-s + (0.809 + 0.587i)17-s + (−0.978 − 0.207i)19-s + (−0.961 + 0.275i)20-s + (−0.719 + 0.694i)22-s + (0.719 − 0.694i)23-s + ⋯
L(s)  = 1  + (−0.848 − 0.529i)2-s + (0.438 + 0.898i)4-s + (−0.173 + 0.984i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (0.669 − 0.743i)10-s + (0.241 − 0.970i)11-s + (0.0348 + 0.999i)13-s + (0.719 − 0.694i)14-s + (−0.615 + 0.788i)16-s + (0.809 + 0.587i)17-s + (−0.978 − 0.207i)19-s + (−0.961 + 0.275i)20-s + (−0.719 + 0.694i)22-s + (0.719 − 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.634 - 0.773i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.634 - 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02401482975 - 0.05077422338i\)
\(L(\frac12)\) \(\approx\) \(0.02401482975 - 0.05077422338i\)
\(L(1)\) \(\approx\) \(0.5989018166 + 0.07071095448i\)
\(L(1)\) \(\approx\) \(0.5989018166 + 0.07071095448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.848 - 0.529i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.241 + 0.970i)T \)
11 \( 1 + (0.241 - 0.970i)T \)
13 \( 1 + (0.0348 + 0.999i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.719 - 0.694i)T \)
29 \( 1 + (-0.848 - 0.529i)T \)
37 \( 1 + T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (-0.882 + 0.469i)T \)
47 \( 1 + (-0.990 - 0.139i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.559 + 0.829i)T \)
83 \( 1 + (0.882 - 0.469i)T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (0.961 - 0.275i)T \)
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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

−22.5473684236368114522832786841, −20.97195925774088686373247399905, −20.436771763841393497696693770163, −19.84311670547004886255411193441, −19.1200373097221340021599088741, −18.020983952871175348912678700761, −17.219789810426816629652341968845, −16.81633015466885673878578610587, −15.99098331559075054633644706458, −15.15492336219340789998585570748, −14.38767250418047252809892023254, −13.22689774325536687998625412380, −12.57062818925390154316720119207, −11.43859190562875648880001298866, −10.492875854610548045824533651025, −9.72002056403810679939206286896, −9.07947698983885902251249973744, −7.89795330581781766025462861644, −7.520488456652312245712674936703, −6.4765972092015021574138900487, −5.36319822268788990800117886695, −4.63285603846781668212333778895, −3.405225280247818905232689407897, −1.80854848937483095160357795684, −0.91927535447610093600007861072, 0.01960985167817519783197011428, 1.55267588476443172759636029670, 2.5942762643431558320413646460, 3.2663833472442803954260118235, 4.30389536550873433591952786959, 6.08005504112846176075145268480, 6.49396070981379385957823818033, 7.6682845216672414385334973988, 8.51648975176082532977946159554, 9.25602237843165601977233031602, 10.12956993719140814789539867018, 11.138124870388845747834435404576, 11.49700735878694821323409635603, 12.47693844476429814759650581086, 13.362069910732948193390745291952, 14.61978344566832404610379576390, 15.15799711996345698055715180403, 16.3594385821725177336359739175, 16.773344361759331578893673236540, 17.950968154297688295428333165251, 18.74379413666219022436187139410, 19.050247526688991332049849186, 19.65733843284515838990335511398, 21.118139081077831623522048099597, 21.48872429922752297881791259873

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