Properties

Label 1-837-837.679-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.773 - 0.634i$
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  + (0.848 − 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

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

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.773 - 0.634i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.773 - 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6001667274 - 1.677388579i\)
\(L(\frac12)\) \(\approx\) \(0.6001667274 - 1.677388579i\)
\(L(1)\) \(\approx\) \(1.247072817 - 0.7661850322i\)
\(L(1)\) \(\approx\) \(1.247072817 - 0.7661850322i\)

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

−22.36698550883591476739177612017, −21.68251274223208795024175797285, −21.1156106141514468342844488000, −20.24220174669480914468227061961, −19.51414080920151272589542182109, −18.19910367146226365181163350252, −17.52457098253103506637569020, −16.619774210832160625726421522660, −15.884053775264192107316154647424, −15.33411257134107283433380010968, −14.402906838323790218435588741855, −13.41755596732662503720231974387, −12.828622674139945291185964628343, −12.027970784667909950616306220047, −11.50608299612870533823259104337, −9.969728360028054698070394319245, −8.956656094036934032896171326, −8.46479448857265335527642808624, −7.23432964419856704492987449809, −6.41613670664378991343017897101, −5.49603237463456393225077276086, −4.69581496851385594732272303025, −4.05312067249090631947316792534, −2.52761225999145088578062904798, −1.8854426693722530173403881582, 0.548001872907781873257100796929, 2.05702059102582819164349660462, 2.98486974145074391005085230700, 3.74430790486437522706828393414, 4.61752174819934901091167269334, 6.07352082089259945634959864495, 6.30504718185914594029944706770, 7.45077334521817225100747208523, 8.50747061078119401078089578037, 10.05368008249223980644553349522, 10.48279842603822427484242683680, 11.013846222014406242019536007833, 12.03925904530564124696576108413, 13.26859514774516840842416000194, 13.45815342103925039056492038692, 14.487759195634143014520470206980, 15.11827995120549839399387800149, 15.99389160354120919373683669121, 16.99343518465951651757869635781, 18.02696847915124881892666903289, 18.823153383995687177241642855844, 19.63802938720948711150504558595, 20.191199335039096634870518888348, 21.22321896187654971661161237833, 21.89821697839199153252710300496

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