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Properties

Label 1-837-837.599-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: $\chi_{837} (599, \cdot )$
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

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

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