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Properties

Label 1-837-837.416-r0-0-0
Degree $1$
Conductor $837$
Sign $0.182 - 0.983i$
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  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (0.939 − 0.342i)5-s + (0.438 − 0.898i)7-s + (−0.913 − 0.406i)8-s + (−0.978 + 0.207i)10-s + (0.438 − 0.898i)11-s + (0.374 − 0.927i)13-s + (−0.559 + 0.829i)14-s + (0.848 + 0.529i)16-s + (−0.809 + 0.587i)17-s + (0.669 + 0.743i)19-s + (0.997 − 0.0697i)20-s + (−0.559 + 0.829i)22-s + (0.559 − 0.829i)23-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (0.939 − 0.342i)5-s + (0.438 − 0.898i)7-s + (−0.913 − 0.406i)8-s + (−0.978 + 0.207i)10-s + (0.438 − 0.898i)11-s + (0.374 − 0.927i)13-s + (−0.559 + 0.829i)14-s + (0.848 + 0.529i)16-s + (−0.809 + 0.587i)17-s + (0.669 + 0.743i)19-s + (0.997 − 0.0697i)20-s + (−0.559 + 0.829i)22-s + (0.559 − 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.182 - 0.983i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (416, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9562684498 - 0.7948547671i\)
\(L(\frac12)\) \(\approx\) \(0.9562684498 - 0.7948547671i\)
\(L(1)\) \(\approx\) \(0.8753129212 - 0.3069987741i\)
\(L(1)\) \(\approx\) \(0.8753129212 - 0.3069987741i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.438 - 0.898i)T \)
11 \( 1 + (0.438 - 0.898i)T \)
13 \( 1 + (0.374 - 0.927i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.559 - 0.829i)T \)
29 \( 1 + (0.990 + 0.139i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 + (0.615 - 0.788i)T \)
47 \( 1 + (-0.0348 + 0.999i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.990 + 0.139i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.241 - 0.970i)T \)
83 \( 1 + (-0.615 + 0.788i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (-0.997 + 0.0697i)T \)
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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

−22.067328961668636937257557307350, −21.42759054594269048358518442000, −20.697036756434270552752357834, −19.826007474421930366666176835481, −18.9009852393094151957462941068, −18.19029805442636667868506872432, −17.63031201010024556432333731594, −17.05050982477535874511953525283, −15.81084101199558144819772657754, −15.318383131648769958076159887505, −14.34504392352959343153021631498, −13.5954005684290653452210184680, −12.25476693462907634021729932053, −11.53117371121377819000543433154, −10.80292600691802113278174259236, −9.61228915393273121723260924026, −9.2665599243935902686090934860, −8.48451483778674373892207604173, −7.10638586982735684359555813827, −6.70332263415044914521679325268, −5.63146687688525791647853736524, −4.75296885835559894866063154968, −3.001542781290811018166175611763, −2.10944143507339992724099999544, −1.43066071926734096758014956450, 0.8516980844935544595593234722, 1.54075132645476784640603851336, 2.79683646756046367675623869865, 3.825975179892459900312419947964, 5.21191307780567413205283481217, 6.205578386047945298725615301415, 6.91318815718268628214288649456, 8.19658071018400912145105004824, 8.57640888284079767417860807261, 9.64292672349004975827495991903, 10.53668877631365548591060212795, 10.87980047150865750369554070317, 12.06654324401430524814863708037, 13.02117772678926334586962888230, 13.84191609620089985960414869358, 14.685935476160028706693513295661, 15.87763991512111944013124232069, 16.57977658280747061639478844535, 17.35042099502232771082513739597, 17.76301544239017187234551387245, 18.68122125956464557653214085259, 19.60255865356353033125494611797, 20.452207481999486978881791256579, 20.83284529911156563768700850159, 21.74844776806792936109261461189

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