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Properties

Label 1-837-837.814-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.814 - 0.580i$
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  + (−0.241 − 0.970i)2-s + (−0.882 + 0.469i)4-s + (−0.939 − 0.342i)5-s + (0.990 − 0.139i)7-s + (0.669 + 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.615 − 0.788i)11-s + (−0.241 + 0.970i)13-s + (−0.374 − 0.927i)14-s + (0.559 − 0.829i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.990 − 0.139i)20-s + (−0.615 + 0.788i)22-s + (0.990 + 0.139i)23-s + ⋯
L(s)  = 1  + (−0.241 − 0.970i)2-s + (−0.882 + 0.469i)4-s + (−0.939 − 0.342i)5-s + (0.990 − 0.139i)7-s + (0.669 + 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.615 − 0.788i)11-s + (−0.241 + 0.970i)13-s + (−0.374 − 0.927i)14-s + (0.559 − 0.829i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.990 − 0.139i)20-s + (−0.615 + 0.788i)22-s + (0.990 + 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.814 - 0.580i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (814, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.814 - 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2483402136 - 0.7759216383i\)
\(L(\frac12)\) \(\approx\) \(0.2483402136 - 0.7759216383i\)
\(L(1)\) \(\approx\) \(0.6267440705 - 0.4329935502i\)
\(L(1)\) \(\approx\) \(0.6267440705 - 0.4329935502i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.241 - 0.970i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.990 - 0.139i)T \)
11 \( 1 + (-0.615 - 0.788i)T \)
13 \( 1 + (-0.241 + 0.970i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.990 + 0.139i)T \)
29 \( 1 + (-0.241 - 0.970i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + (0.961 + 0.275i)T \)
47 \( 1 + (0.438 + 0.898i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.961 - 0.275i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (0.848 - 0.529i)T \)
83 \( 1 + (-0.241 - 0.970i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (-0.615 - 0.788i)T \)
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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

−22.58707450363116353847210518782, −22.080617945996040878063350043213, −20.613059895880615657642989039230, −20.118220270066779645202266374973, −19.02024902867318270438571486503, −18.201782937331616392032833572328, −17.80505561914705789242166296647, −16.8534068127800238045695736122, −15.80408796063825756509498122393, −15.2175071219886912499937446874, −14.79827701812926911799157846902, −13.77656210801180023179131306329, −12.80787710274969222672481389773, −11.88174291432163874280336040637, −10.81843404693936441402474641224, −10.18667927258180680023003241666, −8.897372624095597713910618521543, −8.22554884188064835480437735204, −7.3725445163475565127296514820, −6.9826717351952941901200200352, −5.42713019580436591220756761136, −4.93236619285373788785577911710, −3.9537041831073129950307143922, −2.66634092764813572878150246247, −1.12139625604016800851799044289, 0.4856021576509113703845741341, 1.66384624426170435206354063906, 2.76415530583909382391899382456, 3.8581732421581609887401312411, 4.61021942534763318269534838534, 5.353193716304729291936812644770, 7.09589771899857309133596922268, 7.89818507210145726702662995625, 8.66242703920829915915940743173, 9.31591886317796152040480390076, 10.63857465345556934987393708731, 11.369641191684033142593324589878, 11.63023042265066986494916025024, 12.767234510929094587924528405552, 13.56327419198281352814330631369, 14.3492320924041097723799073156, 15.42548517801555974967218298778, 16.306680984773528019074756078581, 17.21292896230714234941326767761, 17.923222674404481244627030405289, 18.95771453802632486811193581187, 19.30757347680438525572383486046, 20.342388275209501305145061210517, 20.87925731364166351116619827240, 21.60808742696816688554222754549

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