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Properties

Label 1-837-837.106-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.649 + 0.760i$
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  + (−0.615 + 0.788i)2-s + (−0.241 − 0.970i)4-s + (0.766 + 0.642i)5-s + (0.559 + 0.829i)7-s + (0.913 + 0.406i)8-s + (−0.978 + 0.207i)10-s + (−0.559 − 0.829i)11-s + (−0.990 + 0.139i)13-s + (−0.997 − 0.0697i)14-s + (−0.882 + 0.469i)16-s + (0.809 − 0.587i)17-s + (0.669 + 0.743i)19-s + (0.438 − 0.898i)20-s + (0.997 + 0.0697i)22-s + (0.997 + 0.0697i)23-s + ⋯
L(s)  = 1  + (−0.615 + 0.788i)2-s + (−0.241 − 0.970i)4-s + (0.766 + 0.642i)5-s + (0.559 + 0.829i)7-s + (0.913 + 0.406i)8-s + (−0.978 + 0.207i)10-s + (−0.559 − 0.829i)11-s + (−0.990 + 0.139i)13-s + (−0.997 − 0.0697i)14-s + (−0.882 + 0.469i)16-s + (0.809 − 0.587i)17-s + (0.669 + 0.743i)19-s + (0.438 − 0.898i)20-s + (0.997 + 0.0697i)22-s + (0.997 + 0.0697i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.649 + 0.760i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.649 + 0.760i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6983516346 + 1.515988792i\)
\(L(\frac12)\) \(\approx\) \(0.6983516346 + 1.515988792i\)
\(L(1)\) \(\approx\) \(0.8099062256 + 0.4975681268i\)
\(L(1)\) \(\approx\) \(0.8099062256 + 0.4975681268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.615 + 0.788i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.559 + 0.829i)T \)
11 \( 1 + (-0.559 - 0.829i)T \)
13 \( 1 + (-0.990 + 0.139i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.997 + 0.0697i)T \)
29 \( 1 + (0.615 - 0.788i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.882 - 0.469i)T \)
43 \( 1 + (0.374 + 0.927i)T \)
47 \( 1 + (0.848 + 0.529i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (-0.615 - 0.788i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.719 + 0.694i)T \)
83 \( 1 + (0.374 + 0.927i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (0.438 - 0.898i)T \)
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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

−21.40804854923690081238882210639, −20.71617717539536124729113443622, −20.18258288704668224775304578997, −19.45267041836244096146547101369, −18.33927445573179888877348220020, −17.5765711746735386755611352681, −17.14757149339193359030240961248, −16.46230045197170997043573406270, −15.2166212052381745559485604086, −14.10883384700043064636901379250, −13.37486397869646796262216422191, −12.55418503920648804565774933375, −11.953503886186257727421556703232, −10.65939445768618348433612991174, −10.21989728753436608260968977951, −9.420747454091768672661641998, −8.52825126886459648536647855929, −7.57549902500440507830885284320, −6.92567041036903367176068233996, −5.10280653752131379763334942477, −4.79772033449429497485368757552, −3.44707214816235344898827728594, −2.321996094784591568839835158002, −1.44724880216377757388874420790, −0.523962304798109549051997262765, 1.01759321117851720390685198335, 2.187603496345127697514354824360, 3.086587154498226415347870458686, 4.94899278766453672529914598116, 5.47677747642742391764643213325, 6.25392029719381285133112006156, 7.33076943263073900857918480444, 7.99811306406394109835895055004, 9.04113608318839338485246782530, 9.72995504645515662154501120833, 10.535591567551990723047358233190, 11.40077443932849211913725384961, 12.44871919345607037670256398905, 13.90686265055646034429430179428, 14.093939284101737514928598716816, 15.09493040173227660248994270103, 15.72539393599091306961830378860, 16.82555192519879920868292510944, 17.35416732254982271447412929583, 18.444901665366822986125692730926, 18.6003135038810807088489718725, 19.4573091773263789939040884982, 20.81983735714793332473762137334, 21.397112612062117600818191752939, 22.355720023280418613466986179752

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