↑

Properties

Label 1-967-967.7-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

Related objects

Downloads

Learn more

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: $\chi_{967} (7, \cdot )$
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 \( 1 + (-0.900 + 0.433i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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