Properties

Label 1-837-837.524-r1-0-0
Degree $1$
Conductor $837$
Sign $0.458 + 0.888i$
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  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.173 + 0.984i)5-s + (−0.997 + 0.0697i)7-s + (−0.913 − 0.406i)8-s + (0.309 − 0.951i)10-s + (−0.559 − 0.829i)11-s + (−0.615 − 0.788i)13-s + (0.997 + 0.0697i)14-s + (0.848 + 0.529i)16-s + (−0.913 − 0.406i)17-s + (0.309 − 0.951i)19-s + (−0.438 + 0.898i)20-s + (0.438 + 0.898i)22-s + (−0.559 + 0.829i)23-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.173 + 0.984i)5-s + (−0.997 + 0.0697i)7-s + (−0.913 − 0.406i)8-s + (0.309 − 0.951i)10-s + (−0.559 − 0.829i)11-s + (−0.615 − 0.788i)13-s + (0.997 + 0.0697i)14-s + (0.848 + 0.529i)16-s + (−0.913 − 0.406i)17-s + (0.309 − 0.951i)19-s + (−0.438 + 0.898i)20-s + (0.438 + 0.898i)22-s + (−0.559 + 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.458 + 0.888i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2268122629 + 0.1381958243i\)
\(L(\frac12)\) \(\approx\) \(0.2268122629 + 0.1381958243i\)
\(L(1)\) \(\approx\) \(0.4541264413 + 0.01348423063i\)
\(L(1)\) \(\approx\) \(0.4541264413 + 0.01348423063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.997 + 0.0697i)T \)
11 \( 1 + (-0.559 - 0.829i)T \)
13 \( 1 + (-0.615 - 0.788i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.559 + 0.829i)T \)
29 \( 1 + (-0.990 - 0.139i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.0348 - 0.999i)T \)
43 \( 1 + (0.990 + 0.139i)T \)
47 \( 1 + (-0.0348 + 0.999i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (0.615 + 0.788i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.961 - 0.275i)T \)
83 \( 1 + (0.374 + 0.927i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (-0.997 + 0.0697i)T \)
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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

−21.65527354099816117743327863387, −20.546846828393027196156030372140, −20.22964024294154942258743712599, −19.37279306593362228131400888559, −18.68050538091782073729875462922, −17.75746173665736464847452074865, −16.82534874759835038371618359765, −16.431416349530257012812112252332, −15.62251448303691445631539108742, −14.87532359930996142837155491540, −13.60742519324283421434305883164, −12.50635998726956851924280259982, −12.19181731317541303511045526915, −10.98162620243393571044790424857, −9.92095536210029477866411711666, −9.512520471626004141078352164636, −8.597166903680540591338550824672, −7.7524980815283011674903798231, −6.87006765330457247463322577158, −6.0186238295780552737774278608, −4.90031092302284298860229800044, −3.82678964735965908065149833009, −2.42520069307703652575289452728, −1.57852868670644773968988671747, −0.167539393311775800987710758500, 0.45541591941412387514722931722, 2.2742477473588387939176203819, 2.917524095478697224368661618483, 3.70009144148586888471027883162, 5.520976539378821208975624982500, 6.34571076863189356584082233573, 7.25524654854786552086444166314, 7.79154381755307354850355938346, 9.06435757849553054276113308458, 9.656995877553349053333908364534, 10.67644635219392761568200671758, 11.075014166275560520851621676241, 12.11044199032368928395094590418, 13.07398375816981293708487530552, 13.96310566555000237582927728355, 15.32632898638351136524789703175, 15.61672800932672983630355700747, 16.43020416432562643235518627532, 17.61762671559073787242334666995, 18.008215888870671853688694835557, 19.05847074869773917151126682027, 19.422068590161509633524312279302, 20.177296153952308833728827921333, 21.2442057463352658691198478846, 22.20678812249311552809311749975

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