Properties

Label 1-837-837.302-r0-0-0
Degree $1$
Conductor $837$
Sign $0.664 + 0.746i$
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  + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.374 + 0.927i)11-s + (0.719 + 0.694i)13-s + (−0.990 − 0.139i)14-s + (−0.997 − 0.0697i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.615 + 0.788i)20-s + (0.374 + 0.927i)22-s + (−0.615 + 0.788i)23-s + ⋯
L(s)  = 1  + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.374 + 0.927i)11-s + (0.719 + 0.694i)13-s + (−0.990 − 0.139i)14-s + (−0.997 − 0.0697i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.615 + 0.788i)20-s + (0.374 + 0.927i)22-s + (−0.615 + 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.664 + 0.746i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (302, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.664 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8899170097 + 0.3992185244i\)
\(L(\frac12)\) \(\approx\) \(0.8899170097 + 0.3992185244i\)
\(L(1)\) \(\approx\) \(1.028132369 - 0.2253504240i\)
\(L(1)\) \(\approx\) \(1.028132369 - 0.2253504240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.719 - 0.694i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
11 \( 1 + (-0.374 + 0.927i)T \)
13 \( 1 + (0.719 + 0.694i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.615 + 0.788i)T \)
29 \( 1 + (-0.719 + 0.694i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.438 + 0.898i)T \)
43 \( 1 + (0.241 + 0.970i)T \)
47 \( 1 + (-0.559 + 0.829i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.241 - 0.970i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (-0.719 + 0.694i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (-0.374 + 0.927i)T \)
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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

−22.36164574151783133308696498370, −21.33873157333470822241810012652, −20.55166496453716162130570825302, −19.8480813887440005349121789944, −18.67376889431095304703468947115, −18.08862168552560087650608543337, −16.84598840432363042604769003515, −16.11873368650031560830096452102, −15.70040202188666551838722787826, −15.05031747933452182526800000081, −13.77626937459053893454834966641, −13.1972283333692758024349181741, −12.38865445895723646827477718656, −11.7180212894936857773737706087, −10.81325370121231515623450939327, −9.28435480171237479518718472828, −8.52914664525142459313893513754, −7.96444457467034642416249669682, −6.87734738235240463060886715753, −5.78413823028679355951359144150, −5.38268885048487109861458743007, −4.08375253852981760285048482113, −3.42062601457508373962012679896, −2.38708393963371702792206957418, −0.35232912389864701337871904430, 1.32241155684069839067383180479, 2.53911965169096962363179483250, 3.538160741230960739158891547036, 4.12084973674639999588542756715, 5.05613178779522471086874799278, 6.48243632487496401426224936674, 6.89783013056691877012773269128, 7.98873553397001282761167177934, 9.459812801741055731835345386326, 10.00920284648683884038415185899, 11.18210914786829512759640776732, 11.35388347346062432029599791295, 12.57631074252634066997791011540, 13.281537068339945670251435773483, 14.01944930753333409413685887092, 14.91187374907867917767128874110, 15.70371111352100609415925871143, 16.268016238878397847914745619961, 17.76322801050038101834928127683, 18.41753318512343926463315393800, 19.33069618093565099433426800861, 20.011984580857716462162809082630, 20.423991138529849743437459849845, 21.57456537549753958494127967082, 22.42040676405312557812245753748

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