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Properties

Label 1-837-837.146-r0-0-0
Degree $1$
Conductor $837$
Sign $0.671 + 0.741i$
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  + (−0.438 + 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.939 − 0.342i)5-s + (−0.882 + 0.469i)7-s + (0.978 − 0.207i)8-s + (−0.104 + 0.994i)10-s + (−0.882 + 0.469i)11-s + (0.997 + 0.0697i)13-s + (−0.0348 − 0.999i)14-s + (−0.241 + 0.970i)16-s + (0.309 − 0.951i)17-s + (0.913 − 0.406i)19-s + (−0.848 − 0.529i)20-s + (−0.0348 − 0.999i)22-s + (0.0348 + 0.999i)23-s + ⋯
L(s)  = 1  + (−0.438 + 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.939 − 0.342i)5-s + (−0.882 + 0.469i)7-s + (0.978 − 0.207i)8-s + (−0.104 + 0.994i)10-s + (−0.882 + 0.469i)11-s + (0.997 + 0.0697i)13-s + (−0.0348 − 0.999i)14-s + (−0.241 + 0.970i)16-s + (0.309 − 0.951i)17-s + (0.913 − 0.406i)19-s + (−0.848 − 0.529i)20-s + (−0.0348 − 0.999i)22-s + (0.0348 + 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.671 + 0.741i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.671 + 0.741i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076088074 + 0.4773733192i\)
\(L(\frac12)\) \(\approx\) \(1.076088074 + 0.4773733192i\)
\(L(1)\) \(\approx\) \(0.8650043828 + 0.3054346153i\)
\(L(1)\) \(\approx\) \(0.8650043828 + 0.3054346153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.438 + 0.898i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.882 + 0.469i)T \)
11 \( 1 + (-0.882 + 0.469i)T \)
13 \( 1 + (0.997 + 0.0697i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.0348 + 0.999i)T \)
29 \( 1 + (0.438 - 0.898i)T \)
37 \( 1 - T \)
41 \( 1 + (0.241 + 0.970i)T \)
43 \( 1 + (-0.559 - 0.829i)T \)
47 \( 1 + (-0.961 + 0.275i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.438 - 0.898i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.374 + 0.927i)T \)
83 \( 1 + (0.559 + 0.829i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (0.848 + 0.529i)T \)
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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

−21.85903603093964506336542893955, −21.119920559002384629810620994584, −20.555456802206780366550658229228, −19.63410013886796965235765642311, −18.710532175927088890573695655372, −18.301971091946465190739978650525, −17.42694779679961537234120145167, −16.515561351878954653647854962546, −15.96177999138928001842407628779, −14.44562106136767014774525525115, −13.64729361410575528411799075086, −13.08026967695891181645961420708, −12.38459096897643698036577827032, −11.083284967927149700086718358454, −10.39503078607390070120814617267, −10.016283930404362696325365249132, −8.92759624858884959112893873622, −8.18254753637979235083147111908, −7.01005070678348041576049565289, −6.06469217006364597442173800654, −5.07883238355475924595218059882, −3.6042934188466081921395618667, −3.140350581205909386100455228821, −1.99496042293601062825373723328, −0.90124207815201543308697875190, 0.87937321416071157331591115151, 2.1468943298367551928456151182, 3.35878883769648361253348141358, 4.908555241658268019650122489424, 5.49104672956200473503351853313, 6.29676291984103902146620676114, 7.13903688852945496534726491248, 8.15992742595100357353597549002, 9.126160795322736629986352831603, 9.684726742499906447867734904607, 10.31453154690552595912144082628, 11.60106255896143099447759994506, 12.86018519753272010010501705662, 13.50251529764416411694734101536, 14.026585098065490468716990974104, 15.38994015400938153141401800609, 15.830403642496298970819625952792, 16.51873393053213939367186132145, 17.51498625688118683383303217933, 18.18025236957850993825156106334, 18.68941556736554394807397177758, 19.74817530950085516801052615919, 20.65357013001301959572073250984, 21.48910018235348355242935652711, 22.4792088425176582683962999076

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