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Properties

Label 1-967-967.646-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.946 + 0.323i$
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  + (0.538 − 0.842i)2-s + (−0.682 + 0.730i)3-s + (−0.419 − 0.907i)4-s + (−0.983 − 0.181i)5-s + (0.247 + 0.968i)6-s + (−0.934 + 0.356i)7-s + (−0.990 − 0.136i)8-s + (−0.0682 − 0.997i)9-s + (−0.682 + 0.730i)10-s + (−0.0682 − 0.997i)11-s + (0.949 + 0.313i)12-s + (0.829 + 0.557i)13-s + (−0.203 + 0.979i)14-s + (0.803 − 0.595i)15-s + (−0.648 + 0.761i)16-s + (−0.917 + 0.398i)17-s + ⋯
L(s)  = 1  + (0.538 − 0.842i)2-s + (−0.682 + 0.730i)3-s + (−0.419 − 0.907i)4-s + (−0.983 − 0.181i)5-s + (0.247 + 0.968i)6-s + (−0.934 + 0.356i)7-s + (−0.990 − 0.136i)8-s + (−0.0682 − 0.997i)9-s + (−0.682 + 0.730i)10-s + (−0.0682 − 0.997i)11-s + (0.949 + 0.313i)12-s + (0.829 + 0.557i)13-s + (−0.203 + 0.979i)14-s + (0.803 − 0.595i)15-s + (−0.648 + 0.761i)16-s + (−0.917 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.946 + 0.323i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (646, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.946 + 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09478708591 - 0.5693280851i\)
\(L(\frac12)\) \(\approx\) \(-0.09478708591 - 0.5693280851i\)
\(L(1)\) \(\approx\) \(0.6778871032 - 0.3262500010i\)
\(L(1)\) \(\approx\) \(0.6778871032 - 0.3262500010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.538 - 0.842i)T \)
3 \( 1 + (-0.682 + 0.730i)T \)
5 \( 1 + (-0.983 - 0.181i)T \)
7 \( 1 + (-0.934 + 0.356i)T \)
11 \( 1 + (-0.0682 - 0.997i)T \)
13 \( 1 + (0.829 + 0.557i)T \)
17 \( 1 + (-0.917 + 0.398i)T \)
19 \( 1 + (0.974 + 0.225i)T \)
23 \( 1 + (0.576 - 0.816i)T \)
29 \( 1 + (0.0682 - 0.997i)T \)
31 \( 1 + (0.613 - 0.789i)T \)
37 \( 1 + (0.829 + 0.557i)T \)
41 \( 1 + (-0.962 - 0.269i)T \)
43 \( 1 + (-0.113 - 0.993i)T \)
47 \( 1 + (-0.803 + 0.595i)T \)
53 \( 1 + (0.995 + 0.0909i)T \)
59 \( 1 + (0.995 - 0.0909i)T \)
61 \( 1 + (-0.715 + 0.699i)T \)
67 \( 1 + (0.990 - 0.136i)T \)
71 \( 1 + (0.460 - 0.887i)T \)
73 \( 1 + (0.995 - 0.0909i)T \)
79 \( 1 + (-0.995 + 0.0909i)T \)
83 \( 1 + (-0.247 - 0.968i)T \)
89 \( 1 + (-0.377 + 0.926i)T \)
97 \( 1 + T \)
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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

−22.50965571488449066667794017819, −21.57145907038686031891267143306, −20.10365472053676130216009133769, −19.8136931299753791631767250668, −18.48554686857001957217946231153, −18.04395707023771885388042601912, −17.2094333002799268225146030210, −16.1755563741352738650551301914, −15.864343884133721029541140317557, −15.05051689655283291682765838898, −13.88456832609202694090674417862, −13.10570194757776559108649721023, −12.651333243390328223429395170873, −11.73063509689119947621409318146, −11.05018904604270615434821441437, −9.82075831643577441941928635493, −8.6304629182123791695609602472, −7.729055486648418820977465975988, −6.924683003218269679112666729792, −6.70358061742334263442155370618, −5.43534148588244623027506908954, −4.67925705675510144831661602567, −3.60900464863575977753061949106, −2.77802211760032205128639197888, −0.96530514868887263549477603677, 0.17219171666253684505729206236, 0.93420673203333125895047616374, 2.70574993698272016760823596904, 3.604388073920002810430630081253, 4.11004569836189010367760577977, 5.10312763549049830929171191417, 6.081923497383456398489158033478, 6.64091273055065684254724717732, 8.43585992948905786112872844296, 9.072977616209296919838854763253, 9.98292668735444200185928248300, 10.880667204976812196177981295768, 11.52497149725382462417414183591, 12.00378358440376185855912344341, 13.02198074101404283229591107162, 13.70030283907508794687678139973, 14.978526521891189156438297127, 15.575898250158014303469881465088, 16.181580147357180590344629382386, 16.96206616718751675724849872129, 18.40172136853703568283710680443, 18.80635885900722501420959152982, 19.669504032008462847804108166068, 20.4750553301777102324566071923, 21.1604055820334594818491330697

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