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Properties

Label 1-967-967.829-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.0863 - 0.996i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.929 + 0.368i)2-s + (−0.924 − 0.380i)3-s + (0.728 + 0.684i)4-s + (0.906 − 0.422i)5-s + (−0.719 − 0.694i)6-s + (−0.602 + 0.797i)7-s + (0.425 + 0.905i)8-s + (0.710 + 0.703i)9-s + (0.998 − 0.0585i)10-s + (−0.844 + 0.536i)11-s + (−0.413 − 0.910i)12-s + (0.0422 − 0.999i)13-s + (−0.854 + 0.519i)14-s + (−0.998 + 0.0455i)15-s + (0.0617 + 0.998i)16-s + (−0.967 + 0.250i)17-s + ⋯
L(s)  = 1  + (0.929 + 0.368i)2-s + (−0.924 − 0.380i)3-s + (0.728 + 0.684i)4-s + (0.906 − 0.422i)5-s + (−0.719 − 0.694i)6-s + (−0.602 + 0.797i)7-s + (0.425 + 0.905i)8-s + (0.710 + 0.703i)9-s + (0.998 − 0.0585i)10-s + (−0.844 + 0.536i)11-s + (−0.413 − 0.910i)12-s + (0.0422 − 0.999i)13-s + (−0.854 + 0.519i)14-s + (−0.998 + 0.0455i)15-s + (0.0617 + 0.998i)16-s + (−0.967 + 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0863 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0863 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.0863 - 0.996i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.0863 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7864102887 - 0.8574991735i\)
\(L(\frac12)\) \(\approx\) \(0.7864102887 - 0.8574991735i\)
\(L(1)\) \(\approx\) \(1.243247833 + 0.1446726285i\)
\(L(1)\) \(\approx\) \(1.243247833 + 0.1446726285i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.929 + 0.368i)T \)
3 \( 1 + (-0.924 - 0.380i)T \)
5 \( 1 + (0.906 - 0.422i)T \)
7 \( 1 + (-0.602 + 0.797i)T \)
11 \( 1 + (-0.844 + 0.536i)T \)
13 \( 1 + (0.0422 - 0.999i)T \)
17 \( 1 + (-0.967 + 0.250i)T \)
19 \( 1 + (0.971 + 0.238i)T \)
23 \( 1 + (-0.987 - 0.155i)T \)
29 \( 1 + (-0.993 - 0.116i)T \)
31 \( 1 + (-0.927 - 0.374i)T \)
37 \( 1 + (-0.754 - 0.655i)T \)
41 \( 1 + (-0.203 - 0.979i)T \)
43 \( 1 + (0.941 - 0.337i)T \)
47 \( 1 + (0.658 + 0.752i)T \)
53 \( 1 + (0.898 + 0.439i)T \)
59 \( 1 + (0.216 - 0.976i)T \)
61 \( 1 + (0.746 - 0.665i)T \)
67 \( 1 + (-0.787 - 0.615i)T \)
71 \( 1 + (-0.145 + 0.989i)T \)
73 \( 1 + (0.898 - 0.439i)T \)
79 \( 1 + (0.618 - 0.785i)T \)
83 \( 1 + (0.516 - 0.856i)T \)
89 \( 1 + (-0.139 + 0.990i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−22.04390332006417795837029513626, −21.14341572197802625963597462171, −20.53528629650264840459513341573, −19.57131130445181682664503017183, −18.48907239456851338397785558322, −17.95960082119435407254315617711, −16.72007795563259813513769187448, −16.26920849710742873243298523855, −15.49221684166981510207794184253, −14.42749747494455456059595804950, −13.47671554658183433923743769074, −13.29782606485617759958787984280, −12.09180014871244345861889605750, −11.21287538934154221013034894016, −10.670899200505550178366869779135, −9.92578893161391114039537883600, −9.252065545848395233062057489380, −7.23586930735624374536117602375, −6.73813927556977534277821591333, −5.85034689352491170004012024256, −5.22946240680119507215030260334, −4.19335828659136146952382694412, −3.38199518231644397648828163190, −2.257384667699683183547340451598, −1.121945389132814069853771356529, 0.18884026048344878294764530366, 1.90684715962087639670312648624, 2.46517833073061121111595098487, 3.86846596656401954636999164473, 5.137243292219381658257875476715, 5.55572063808534855408372531909, 6.1076842857720400691601412033, 7.131763169402113753953672228404, 7.951204938168293497867469649798, 9.16828776411439853758885846489, 10.243724677697816294184888892574, 10.97076433611283800981708913368, 12.17439440136750166281487425143, 12.61842355120371788554010897397, 13.178281818861309265289798816477, 13.91729343384066671179758521149, 15.19453489065727247761492700128, 15.84622894570633334306259277511, 16.40760181782969756508312376725, 17.53381969164593023052867494682, 17.84083379942707340698636578240, 18.76547591054095821097603980939, 20.12894376568251399028486299687, 20.69426575800885387708004259374, 21.70160032731691524012238990171

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