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Properties

Label 1-967-967.47-r1-0-0
Degree $1$
Conductor $967$
Sign $0.0351 - 0.999i$
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  + (−0.235 + 0.971i)2-s + (0.442 + 0.896i)3-s + (−0.889 − 0.457i)4-s + (−0.969 + 0.244i)5-s + (−0.975 + 0.219i)6-s + (0.658 − 0.752i)7-s + (0.653 − 0.756i)8-s + (−0.608 + 0.793i)9-s + (−0.00975 − 0.999i)10-s + (0.909 + 0.416i)11-s + (0.0162 − 0.999i)12-s + (0.0941 − 0.995i)13-s + (0.576 + 0.816i)14-s + (−0.648 − 0.761i)15-s + (0.581 + 0.813i)16-s + (−0.844 + 0.536i)17-s + ⋯
L(s)  = 1  + (−0.235 + 0.971i)2-s + (0.442 + 0.896i)3-s + (−0.889 − 0.457i)4-s + (−0.969 + 0.244i)5-s + (−0.975 + 0.219i)6-s + (0.658 − 0.752i)7-s + (0.653 − 0.756i)8-s + (−0.608 + 0.793i)9-s + (−0.00975 − 0.999i)10-s + (0.909 + 0.416i)11-s + (0.0162 − 0.999i)12-s + (0.0941 − 0.995i)13-s + (0.576 + 0.816i)14-s + (−0.648 − 0.761i)15-s + (0.581 + 0.813i)16-s + (−0.844 + 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0351 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0351 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0351 - 0.999i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.0351 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1607218042 + 0.1551736996i\)
\(L(\frac12)\) \(\approx\) \(-0.1607218042 + 0.1551736996i\)
\(L(1)\) \(\approx\) \(0.6145407384 + 0.5201746671i\)
\(L(1)\) \(\approx\) \(0.6145407384 + 0.5201746671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.235 + 0.971i)T \)
3 \( 1 + (0.442 + 0.896i)T \)
5 \( 1 + (-0.969 + 0.244i)T \)
7 \( 1 + (0.658 - 0.752i)T \)
11 \( 1 + (0.909 + 0.416i)T \)
13 \( 1 + (0.0941 - 0.995i)T \)
17 \( 1 + (-0.844 + 0.536i)T \)
19 \( 1 + (-0.304 - 0.952i)T \)
23 \( 1 + (0.477 + 0.878i)T \)
29 \( 1 + (0.999 + 0.0195i)T \)
31 \( 1 + (-0.993 + 0.110i)T \)
37 \( 1 + (0.837 - 0.547i)T \)
41 \( 1 + (-0.682 + 0.730i)T \)
43 \( 1 + (0.395 + 0.918i)T \)
47 \( 1 + (0.999 - 0.0325i)T \)
53 \( 1 + (-0.715 - 0.699i)T \)
59 \( 1 + (-0.992 - 0.123i)T \)
61 \( 1 + (0.291 + 0.956i)T \)
67 \( 1 + (-0.592 + 0.805i)T \)
71 \( 1 + (-0.724 + 0.689i)T \)
73 \( 1 + (-0.715 + 0.699i)T \)
79 \( 1 + (-0.947 + 0.319i)T \)
83 \( 1 + (-0.00325 + 0.999i)T \)
89 \( 1 + (-0.401 - 0.915i)T \)
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

−20.55420095935603607926967297739, −20.26155932214973974920236530641, −19.163962310127489390122587230864, −18.88969467395599099310752957574, −18.2302190409971655746321441259, −17.21018296249077890066101944248, −16.465273446082231212909948135937, −15.17507276358971146672434178572, −14.33262102670525842823448748622, −13.77184887970269386636767074028, −12.57874078145209535051248532220, −12.081505706257399839605016085187, −11.54105705785139397220272891019, −10.80031073098726424702357211850, −9.08508078205919409738893924792, −8.90889646512524108747368282736, −8.127972477206938953949007198827, −7.21965126892263294111503226493, −6.14805860458320097020713491266, −4.73135511765796718712628360352, −3.98278438627375771171520491817, −2.97022332774787575407798206536, −1.99091503037422365919665875606, −1.19344918780910463664440826955, −0.05497924992097180316590262075, 1.24541657183841617202915761176, 3.01257473570074214348218276014, 4.20205104737059804260744415853, 4.36785046465607748985450042959, 5.48180556314594079410611533560, 6.78125930426388692116795326889, 7.4932690424773815184305388272, 8.28907737198361803546897009839, 8.918236340393738549163041552367, 9.90291942694024403017404617835, 10.82284857936154706674415770494, 11.3171295983537373609305519271, 12.834263003866466104523758362550, 13.701634535047460091144143703797, 14.703314114660586132762454108373, 14.97401844197058854961469412053, 15.71303623651681653110299139097, 16.474065517480865220176490015396, 17.41819108118319204397136589815, 17.800528253567613327594787081063, 19.20652177199095761408211609819, 19.880203884430836975570111564051, 20.19012368902150495391158940680, 21.61502989572779084065298576195, 22.21171805364127092004570062349

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