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Properties

Label 1-837-837.508-r1-0-0
Degree $1$
Conductor $837$
Sign $0.983 - 0.182i$
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  + (0.990 − 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

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

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.983 - 0.182i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.983 - 0.182i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.018300301 - 0.3703957800i\)
\(L(\frac12)\) \(\approx\) \(4.018300301 - 0.3703957800i\)
\(L(1)\) \(\approx\) \(1.924401036 - 0.1584295538i\)
\(L(1)\) \(\approx\) \(1.924401036 - 0.1584295538i\)

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

−22.322628834820760074881643188077, −20.983177511937649491547791868686, −20.45604222224144027318519195789, −19.98349243919972220078488025553, −18.88671773455460096207394865492, −17.8960233454259354881480171076, −17.00455546771036356192613512593, −15.99454368184217725783651607683, −15.52189960536683640183652556362, −14.60957943641814468635267312190, −14.00278301738545072823211420175, −13.0533995237346229203055858875, −12.19954460562207470310439927964, −11.531821936824106085905457428913, −10.62797273339068621244267646361, −9.992856260409067458046409083645, −8.2196507742764900576165285099, −7.40279701253985802042240552545, −7.259997509409751610376013181580, −5.715537589355050548881219733178, −5.007590315726312022703591833516, −3.84528415703653970765693423549, −3.48916862351246559131412440755, −2.16122928147025617522688231744, −0.83463771189819087872151637998, 0.86585427440909683796739201704, 2.0790395228507937471071997608, 3.16960645569461335166546555379, 3.97565913718840731159101578603, 4.96088934154713252552506864867, 5.67503825994124515368640625952, 6.646513387865461953262666101666, 7.79224691878957078456394921524, 8.44038525209279664406707192565, 9.53434297549591204728200082279, 11.04848627417567058107274739542, 11.28195623158601522115337809950, 12.243247583602988241562976747756, 12.79463488211802326051742653588, 13.91468324984344942010470741981, 14.5911211120647230861653209218, 15.44813249543058871142695417312, 16.13318031413937924386793655015, 16.643717959755539639141797222370, 18.13733453498593345060130821213, 19.062508458818583591972810804355, 19.444535590553290319015975875435, 20.70648819316100169772767586707, 21.05757002822247969173623855933, 21.95009575702822228181849166091

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