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Properties

Label 1-319-319.118-r0-0-0
Degree $1$
Conductor $319$
Sign $0.993 - 0.114i$
Analytic cond. $1.48142$
Root an. cond. $1.48142$
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  + (−0.512 + 0.858i)2-s + (−0.880 − 0.473i)3-s + (−0.473 − 0.880i)4-s + (0.995 − 0.0896i)5-s + (0.858 − 0.512i)6-s + (−0.983 − 0.178i)7-s + (0.998 + 0.0448i)8-s + (0.550 + 0.834i)9-s + (−0.433 + 0.900i)10-s + i·12-s + (0.936 − 0.351i)13-s + (0.657 − 0.753i)14-s + (−0.919 − 0.393i)15-s + (−0.550 + 0.834i)16-s + (−0.951 + 0.309i)17-s + (−0.998 + 0.0448i)18-s + ⋯
L(s)  = 1  + (−0.512 + 0.858i)2-s + (−0.880 − 0.473i)3-s + (−0.473 − 0.880i)4-s + (0.995 − 0.0896i)5-s + (0.858 − 0.512i)6-s + (−0.983 − 0.178i)7-s + (0.998 + 0.0448i)8-s + (0.550 + 0.834i)9-s + (−0.433 + 0.900i)10-s + i·12-s + (0.936 − 0.351i)13-s + (0.657 − 0.753i)14-s + (−0.919 − 0.393i)15-s + (−0.550 + 0.834i)16-s + (−0.951 + 0.309i)17-s + (−0.998 + 0.0448i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(1.48142\)
Root analytic conductor: \(1.48142\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (0:\ ),\ 0.993 - 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7209818794 - 0.04133364924i\)
\(L(\frac12)\) \(\approx\) \(0.7209818794 - 0.04133364924i\)
\(L(1)\) \(\approx\) \(0.6813320670 + 0.07193994914i\)
\(L(1)\) \(\approx\) \(0.6813320670 + 0.07193994914i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.512 + 0.858i)T \)
3 \( 1 + (-0.880 - 0.473i)T \)
5 \( 1 + (0.995 - 0.0896i)T \)
7 \( 1 + (-0.983 - 0.178i)T \)
13 \( 1 + (0.936 - 0.351i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (0.178 + 0.983i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (0.512 - 0.858i)T \)
37 \( 1 + (-0.266 - 0.963i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
43 \( 1 + (0.974 - 0.222i)T \)
47 \( 1 + (0.266 - 0.963i)T \)
53 \( 1 + (0.858 + 0.512i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.722 - 0.691i)T \)
67 \( 1 + (-0.623 + 0.781i)T \)
71 \( 1 + (0.550 - 0.834i)T \)
73 \( 1 + (0.919 + 0.393i)T \)
79 \( 1 + (0.834 - 0.550i)T \)
83 \( 1 + (-0.134 + 0.990i)T \)
89 \( 1 + (0.974 + 0.222i)T \)
97 \( 1 + (-0.722 - 0.691i)T \)
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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

−25.699729929915453066852161892081, −24.227532845742998897029262545064, −22.90867709724511144823206943308, −22.31010883521965877407452905605, −21.579581935791289216824035018794, −20.91649234244197402268123771359, −19.82434816765454487322139785035, −18.76216739350814918290723695402, −17.86214589987144439172097934467, −17.36665350850619302713543100298, −16.2662093828964280015831991848, −15.60767066996553891235377354476, −13.74493633186101788947356431631, −13.14970234824310301508771467111, −12.10074418382168203118104604835, −11.10406013262582564820528267326, −10.38385831951331859096681194737, −9.39254242305258227788305918355, −8.966623143297475051760242111788, −7.00700742525448635494221724225, −6.15659046638504368045758643199, −4.94342626460580805719337988424, −3.71852019463653827872271339080, −2.55634285317163455692317378889, −1.0916459832585335036593782000, 0.77711457949007877843152138372, 2.11416713440650926073142137372, 4.21312984870262978343595500598, 5.61935594919178001939786679237, 6.16607938039224561730437678067, 6.84817519496329285304759814805, 8.132027332881791779303432915454, 9.27038142173717061529090284896, 10.24982840530309586362270994345, 10.87414125022182104676843410606, 12.56485450058923792278516037161, 13.300321983794839595963347746014, 14.04833141581648580037616937096, 15.48741278407194519653194606379, 16.38835879937828495032591233926, 16.93497042182997741607201093407, 17.92562222886164166766244583529, 18.44941021408941976711975158688, 19.398219737989084687965816710220, 20.59412636425054645111583379819, 22.00497239267761539539194789224, 22.6950870799304953217768621943, 23.33797434908554979730264003531, 24.54043037432740447273427903637, 24.96674105427182011390058685595

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