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Properties

Label 1-837-837.268-r0-0-0
Degree $1$
Conductor $837$
Sign $0.617 + 0.786i$
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  + (0.961 − 0.275i)2-s + (0.848 − 0.529i)4-s + (0.766 + 0.642i)5-s + (−0.615 + 0.788i)7-s + (0.669 − 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.615 + 0.788i)11-s + (−0.719 + 0.694i)13-s + (−0.374 + 0.927i)14-s + (0.438 − 0.898i)16-s + (0.309 + 0.951i)17-s + (−0.104 − 0.994i)19-s + (0.990 + 0.139i)20-s + (−0.374 + 0.927i)22-s + (−0.374 + 0.927i)23-s + ⋯
L(s)  = 1  + (0.961 − 0.275i)2-s + (0.848 − 0.529i)4-s + (0.766 + 0.642i)5-s + (−0.615 + 0.788i)7-s + (0.669 − 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.615 + 0.788i)11-s + (−0.719 + 0.694i)13-s + (−0.374 + 0.927i)14-s + (0.438 − 0.898i)16-s + (0.309 + 0.951i)17-s + (−0.104 − 0.994i)19-s + (0.990 + 0.139i)20-s + (−0.374 + 0.927i)22-s + (−0.374 + 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.617 + 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.395154307 + 1.164378532i\)
\(L(\frac12)\) \(\approx\) \(2.395154307 + 1.164378532i\)
\(L(1)\) \(\approx\) \(1.871135543 + 0.2648454681i\)
\(L(1)\) \(\approx\) \(1.871135543 + 0.2648454681i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.961 - 0.275i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.615 + 0.788i)T \)
11 \( 1 + (-0.615 + 0.788i)T \)
13 \( 1 + (-0.719 + 0.694i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.374 + 0.927i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
37 \( 1 + T \)
41 \( 1 + (0.438 + 0.898i)T \)
43 \( 1 + (-0.241 + 0.970i)T \)
47 \( 1 + (-0.997 + 0.0697i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (0.961 + 0.275i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.882 + 0.469i)T \)
83 \( 1 + (-0.241 + 0.970i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (0.990 + 0.139i)T \)
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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

−22.116964647267309911789060336898, −21.27194944289371038083563038197, −20.51865617181003349365472555017, −20.07687831577115012337768933100, −18.93301791578807418904524782875, −17.78812789418480244359739616255, −16.92619577231784317875026439182, −16.30844594722640342998296886244, −15.797847005030296968267390447187, −14.33283691193365569702709923674, −14.06135296034070356750298345774, −12.980875921932327174683496423307, −12.70274019657986652292166239670, −11.6360426209908151823015847601, −10.38761182300616538808731892428, −9.98886413595242240578485498808, −8.560987450677962656382808689496, −7.74528910921760683432372193149, −6.76169915587191668723588666946, −5.855821681787514697249726646465, −5.19010184228145047361295015392, −4.2549081975031331531782882278, −3.14413385401201526769354361837, −2.354489162559600704370801107684, −0.82787359414554074249544908855, 1.73920310560439637162459684889, 2.47891661718074162183251372408, 3.18974162866160650857372517657, 4.50426761743065472988719284722, 5.3365694228250296711903802375, 6.27442780951753580902558180281, 6.79740067334958235880428121157, 7.92179537514520449117422350160, 9.66324382674389754079352793825, 9.74022924750402607514781937271, 10.93310700976237668008521340575, 11.75188094820715345248982863527, 12.718133735042736368941147359528, 13.180899067058913826845684852418, 14.19494945198419843175420723581, 14.93265115159744751704835217175, 15.489510580064892333395130471786, 16.45829854308733059181419350660, 17.548085253158785620363961991331, 18.3297909746577787962480487840, 19.383616967403463831615511543311, 19.71598675239722601667608865351, 21.119890011422531507539760859174, 21.5568549589522474720961416169, 22.07709777095797685282505364894

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