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Properties

Label 1-837-837.131-r1-0-0
Degree $1$
Conductor $837$
Sign $0.782 - 0.623i$
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  + (0.997 + 0.0697i)2-s + (0.990 + 0.139i)4-s + (−0.173 − 0.984i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (−0.104 − 0.994i)10-s + (−0.848 + 0.529i)11-s + (0.559 + 0.829i)13-s + (0.882 − 0.469i)14-s + (0.961 + 0.275i)16-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s + (−0.0348 − 0.999i)20-s + (−0.882 + 0.469i)22-s + (0.882 − 0.469i)23-s + ⋯
L(s)  = 1  + (0.997 + 0.0697i)2-s + (0.990 + 0.139i)4-s + (−0.173 − 0.984i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (−0.104 − 0.994i)10-s + (−0.848 + 0.529i)11-s + (0.559 + 0.829i)13-s + (0.882 − 0.469i)14-s + (0.961 + 0.275i)16-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s + (−0.0348 − 0.999i)20-s + (−0.882 + 0.469i)22-s + (0.882 − 0.469i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.782 - 0.623i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.782 - 0.623i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.847196079 - 1.694586461i\)
\(L(\frac12)\) \(\approx\) \(4.847196079 - 1.694586461i\)
\(L(1)\) \(\approx\) \(2.273995879 - 0.3555168342i\)
\(L(1)\) \(\approx\) \(2.273995879 - 0.3555168342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.848 - 0.529i)T \)
11 \( 1 + (-0.848 + 0.529i)T \)
13 \( 1 + (0.559 + 0.829i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.882 - 0.469i)T \)
29 \( 1 + (0.997 + 0.0697i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (0.438 + 0.898i)T \)
47 \( 1 + (0.719 - 0.694i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (0.997 - 0.0697i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (-0.438 - 0.898i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (0.0348 + 0.999i)T \)
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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

−22.04914731834406289870928813571, −21.36180828310515038421023312079, −20.72266252267823032251318571340, −19.73418863591266523056891856405, −18.89468402636976816718745245185, −18.104403751546629792722234125805, −17.31397688814587676275551822768, −15.901708757269407425665547005596, −15.43104927881289167889378003705, −14.81748316577996210721165941786, −13.90540503338863272535215761347, −13.2560706843293794151390339461, −12.27476255372231571727045851409, −11.25712972220395307217146608875, −10.93723528434476911990134916826, −10.06938411770252756421057553516, −8.46542630680310439758018556296, −7.7602779603505772866580408101, −6.83269290204174692494916940864, −5.78076119631161289282211354429, −5.26277832124516671885213900571, −4.04458473625802196863730204372, −3.04428435676739507015209526768, −2.45627988389702232854732781239, −1.11985060508430339940994364114, 0.8944608140414140450074726605, 1.82573323185217122039690547085, 3.01121596904315556152456854948, 4.30678924402014286998286237547, 4.74534528940574617736218652472, 5.48770828481491929483785090639, 6.75755274331591844031422473645, 7.60212894673363698600861027093, 8.32910596539102360294188305792, 9.49885656073894562781619893727, 10.63329599369678794966057082207, 11.48711015639102514237494721950, 12.08765907647073903507212877850, 13.08042518342283428157488604382, 13.67963886962069237069366854458, 14.41994482015880518856435573715, 15.44734158166318406781302364797, 16.146901623308526397689493834265, 16.75493469848217966379413884285, 17.731000088658161706844119437151, 18.68305862170846343751693453640, 19.97496104172528122802135418883, 20.4849582986782008901123098857, 20.97554212321769195054559073904, 21.695279147805787288886459067640

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