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Properties

Label 1-837-837.38-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.262 - 0.964i$
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  + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.173 + 0.984i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (−0.809 + 0.587i)10-s + (0.615 − 0.788i)11-s + (0.961 + 0.275i)13-s + (0.374 − 0.927i)14-s + (−0.997 + 0.0697i)16-s + (−0.669 + 0.743i)17-s + (−0.809 + 0.587i)19-s + (−0.990 − 0.139i)20-s + (0.990 − 0.139i)22-s + (0.615 + 0.788i)23-s + ⋯
L(s)  = 1  + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.173 + 0.984i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (−0.809 + 0.587i)10-s + (0.615 − 0.788i)11-s + (0.961 + 0.275i)13-s + (0.374 − 0.927i)14-s + (−0.997 + 0.0697i)16-s + (−0.669 + 0.743i)17-s + (−0.809 + 0.587i)19-s + (−0.990 − 0.139i)20-s + (0.990 − 0.139i)22-s + (0.615 + 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.262 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3300079349 + 0.4316797795i\)
\(L(\frac12)\) \(\approx\) \(-0.3300079349 + 0.4316797795i\)
\(L(1)\) \(\approx\) \(0.9819724513 + 0.6882693076i\)
\(L(1)\) \(\approx\) \(0.9819724513 + 0.6882693076i\)

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 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.374 - 0.927i)T \)
11 \( 1 + (0.615 - 0.788i)T \)
13 \( 1 + (0.961 + 0.275i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.615 + 0.788i)T \)
29 \( 1 + (0.719 + 0.694i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.559 + 0.829i)T \)
43 \( 1 + (-0.719 - 0.694i)T \)
47 \( 1 + (-0.559 - 0.829i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (-0.961 - 0.275i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (0.241 - 0.970i)T \)
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

−21.19838544019260898565261690399, −20.72595956220809366438698535292, −19.89488737706650188240887859074, −19.27000477864311297725763221061, −18.358360395831581883195319367429, −17.491212801590440983110579159815, −16.33851318296358933006526429500, −15.47304207081977695639200831746, −15.08824710347891976290409751018, −13.79945808200970507539146816544, −13.09054716490014788338806923826, −12.42008961279764619237103064597, −11.789333022206834840109772579633, −10.94668352846385176593155426980, −9.77974006718593040962502069783, −9.06084716795790101159403809966, −8.4164862951836889556508741577, −6.74319972037437877577554562038, −6.100179893164373196577677808611, −4.91162007834368488769931972371, −4.48442388727685317715351373675, −3.28431578805163630842582294018, −2.27020859181796526803196405524, −1.27871333202463009909333968107, −0.08887651053160716033665736084, 1.68856330739048309998016298891, 3.31444773306736082974764054850, 3.5883199467854882493726018862, 4.56936460251753942006726873643, 6.079639727012446088067822010311, 6.44369460587484864948981972243, 7.238808543244756478676578089180, 8.22525639580302168141917975018, 9.05340452976412206703482943547, 10.5175847234369552736735611048, 11.03534284021506935355185112374, 11.95963569636048036493922498641, 13.12561864971518556651840319928, 13.70734289381245833001742309714, 14.385025378796459640967040469069, 15.1896796838091246281977673355, 16.003497271813511653831566069432, 16.78995459860788836474000258642, 17.46610616938622031673147593835, 18.45869222692751237024208888092, 19.3474495121181783884666148506, 20.11611738142266029021025543772, 21.343618828844264050486739453694, 21.74192987617946419725823856357, 22.688913204992178751887569294043

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