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Properties

Label 1-837-837.740-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: $\chi_{837} (740, \cdot )$
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

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

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