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Properties

Label 1-967-967.134-r1-0-0
Degree $1$
Conductor $967$
Sign $0.304 - 0.952i$
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  + (−0.383 − 0.923i)2-s + (−0.938 − 0.344i)3-s + (−0.705 + 0.708i)4-s + (0.566 + 0.824i)5-s + (0.0422 + 0.999i)6-s + (−0.505 + 0.862i)7-s + (0.924 + 0.380i)8-s + (0.763 + 0.646i)9-s + (0.544 − 0.838i)10-s + (0.981 − 0.193i)11-s + (0.906 − 0.422i)12-s + (0.322 + 0.946i)13-s + (0.990 + 0.136i)14-s + (−0.247 − 0.968i)15-s + (−0.00325 − 0.999i)16-s + (0.389 − 0.921i)17-s + ⋯
L(s)  = 1  + (−0.383 − 0.923i)2-s + (−0.938 − 0.344i)3-s + (−0.705 + 0.708i)4-s + (0.566 + 0.824i)5-s + (0.0422 + 0.999i)6-s + (−0.505 + 0.862i)7-s + (0.924 + 0.380i)8-s + (0.763 + 0.646i)9-s + (0.544 − 0.838i)10-s + (0.981 − 0.193i)11-s + (0.906 − 0.422i)12-s + (0.322 + 0.946i)13-s + (0.990 + 0.136i)14-s + (−0.247 − 0.968i)15-s + (−0.00325 − 0.999i)16-s + (0.389 − 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.304 - 0.952i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008077853 - 0.7356772559i\)
\(L(\frac12)\) \(\approx\) \(1.008077853 - 0.7356772559i\)
\(L(1)\) \(\approx\) \(0.7245479284 - 0.2160273372i\)
\(L(1)\) \(\approx\) \(0.7245479284 - 0.2160273372i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.383 - 0.923i)T \)
3 \( 1 + (-0.938 - 0.344i)T \)
5 \( 1 + (0.566 + 0.824i)T \)
7 \( 1 + (-0.505 + 0.862i)T \)
11 \( 1 + (0.981 - 0.193i)T \)
13 \( 1 + (0.322 + 0.946i)T \)
17 \( 1 + (0.389 - 0.921i)T \)
19 \( 1 + (-0.413 + 0.910i)T \)
23 \( 1 + (0.883 - 0.468i)T \)
29 \( 1 + (0.407 - 0.913i)T \)
31 \( 1 + (0.692 - 0.721i)T \)
37 \( 1 + (0.120 - 0.992i)T \)
41 \( 1 + (0.334 + 0.942i)T \)
43 \( 1 + (-0.100 - 0.994i)T \)
47 \( 1 + (-0.643 - 0.765i)T \)
53 \( 1 + (0.803 - 0.595i)T \)
59 \( 1 + (-0.465 - 0.884i)T \)
61 \( 1 + (-0.648 + 0.761i)T \)
67 \( 1 + (-0.279 + 0.960i)T \)
71 \( 1 + (-0.608 - 0.793i)T \)
73 \( 1 + (0.803 + 0.595i)T \)
79 \( 1 + (0.759 - 0.651i)T \)
83 \( 1 + (0.754 - 0.655i)T \)
89 \( 1 + (0.971 - 0.238i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−21.9014806002184402144939906426, −21.06543002390509462554775031491, −19.92810362366445189987250249, −19.41994844596437503694055817641, −18.11128924455131733625814434244, −17.3353175570448564788944108954, −17.12507676483406158882134681889, −16.39651794115868296003283187169, −15.619913910957357040002718573404, −14.825497945183249076284775324932, −13.71080414087588970473394590275, −13.0043284273230769698573260510, −12.30869980630326315335116129734, −10.86013951966862981254631466864, −10.3300150150052408397316926903, −9.4776588089721334959316789171, −8.83451728597226545629540273434, −7.69390677530533730044567137320, −6.57735512306787506812451891797, −6.23657168352226955699026393223, −5.14015643395112183202341120778, −4.54815049491364773478312061435, −3.53136580998456744305132570997, −1.21398578157678676221246142142, −0.921384738956419089044588479499, 0.483058069905798049079251999719, 1.683847880001285047776035087145, 2.43249048212874540666665031303, 3.5286963573614162605376473075, 4.59860309109699480447415303921, 5.80077543079567273428747756216, 6.46818275509695011015050616128, 7.31289842255502478336803886917, 8.60202417624868692217367719098, 9.52271279685354851046678911951, 10.06835204681988761034813598318, 11.10761966003748803354115813536, 11.69648700118336890739021241914, 12.24821625224100205755788183574, 13.2683740757839218250104336431, 13.94944492746250635781194391546, 14.93618937479365739247943258706, 16.33094489592978731753782120211, 16.78426082701483795727543225591, 17.70255015560526065407410923460, 18.42854315426938016118202914040, 18.97399684228679946836339640464, 19.35559983992208821524686880752, 20.91228198312158855228575870781, 21.44256476636089464910398010268

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