↑

Properties

Label 1-967-967.58-r1-0-0
Degree $1$
Conductor $967$
Sign $0.379 - 0.925i$
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  + (−0.895 + 0.445i)2-s + (−0.00975 + 0.999i)3-s + (0.602 − 0.797i)4-s + (0.767 + 0.641i)5-s + (−0.436 − 0.899i)6-s + (−0.880 + 0.474i)7-s + (−0.184 + 0.982i)8-s + (−0.999 − 0.0195i)9-s + (−0.972 − 0.232i)10-s + (−0.638 + 0.769i)11-s + (0.791 + 0.610i)12-s + (0.347 − 0.937i)13-s + (0.576 − 0.816i)14-s + (−0.648 + 0.761i)15-s + (−0.272 − 0.962i)16-s + (0.527 + 0.849i)17-s + ⋯
L(s)  = 1  + (−0.895 + 0.445i)2-s + (−0.00975 + 0.999i)3-s + (0.602 − 0.797i)4-s + (0.767 + 0.641i)5-s + (−0.436 − 0.899i)6-s + (−0.880 + 0.474i)7-s + (−0.184 + 0.982i)8-s + (−0.999 − 0.0195i)9-s + (−0.972 − 0.232i)10-s + (−0.638 + 0.769i)11-s + (0.791 + 0.610i)12-s + (0.347 − 0.937i)13-s + (0.576 − 0.816i)14-s + (−0.648 + 0.761i)15-s + (−0.272 − 0.962i)16-s + (0.527 + 0.849i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3616578055 + 0.2425741320i\)
\(L(\frac12)\) \(\approx\) \(-0.3616578055 + 0.2425741320i\)
\(L(1)\) \(\approx\) \(0.4261037513 + 0.4637821891i\)
\(L(1)\) \(\approx\) \(0.4261037513 + 0.4637821891i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.895 + 0.445i)T \)
3 \( 1 + (-0.00975 + 0.999i)T \)
5 \( 1 + (0.767 + 0.641i)T \)
7 \( 1 + (-0.880 + 0.474i)T \)
11 \( 1 + (-0.638 + 0.769i)T \)
13 \( 1 + (0.347 - 0.937i)T \)
17 \( 1 + (0.527 + 0.849i)T \)
19 \( 1 + (0.996 + 0.0844i)T \)
23 \( 1 + (-0.811 + 0.584i)T \)
29 \( 1 + (-0.892 + 0.451i)T \)
31 \( 1 + (0.847 + 0.530i)T \)
37 \( 1 + (-0.719 + 0.694i)T \)
41 \( 1 + (-0.682 - 0.730i)T \)
43 \( 1 + (-0.754 + 0.655i)T \)
47 \( 1 + (-0.254 + 0.967i)T \)
53 \( 1 + (-0.715 + 0.699i)T \)
59 \( 1 + (0.341 + 0.940i)T \)
61 \( 1 + (0.291 - 0.956i)T \)
67 \( 1 + (0.883 + 0.468i)T \)
71 \( 1 + (0.833 - 0.552i)T \)
73 \( 1 + (-0.715 - 0.699i)T \)
79 \( 1 + (-0.840 + 0.541i)T \)
83 \( 1 + (-0.430 + 0.902i)T \)
89 \( 1 + (-0.0357 - 0.999i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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

−20.50402048701391963770231387400, −20.233420919883994642566604855535, −19.0610445110760753877076443092, −18.67873502325111846830845128582, −17.95105396618063789052104155705, −17.02151153047332419082082854128, −16.44345819821558460780236426576, −15.91978918429804329765024636398, −14.08823071029562653402847011250, −13.51133865147279827374435670793, −12.947877147621893368578261864311, −11.978052531744487435072028672881, −11.39043743328128050442965034225, −10.17073100508737815907543480494, −9.561491959502024466645622037911, −8.68947224382534727571128156114, −7.94044010008951955211531218398, −6.97107923328959077788204925873, −6.29038274280304951149882172731, −5.34115388004727440852911291942, −3.70313531074125317629315294085, −2.74677674075318599001325547735, −1.85904836782845258091388098111, −0.83657152294292045047713708436, −0.1502890957393505743928908046, 1.58264524156391753680198314847, 2.78130709018825877570638239180, 3.41635361916217799679867153035, 5.18104803325765225121176527269, 5.69556746646672616836174248412, 6.456870085781589146732796773085, 7.57213797486706702047487963725, 8.47995906760824663502565345798, 9.488950867352263956045452493521, 10.05685563678158008123849741405, 10.36837844036066182699896065001, 11.40889683559125679652845214614, 12.53265275197858542535663779538, 13.70561672379027625127466218336, 14.56465143626033783855262269646, 15.52916874171832100155457779037, 15.60155024584089951882911132629, 16.71281696452486242157937011142, 17.4858347311242697616564754066, 18.119456444844755651320665462483, 18.88202836477692491130739239086, 19.86014613490066205634414506568, 20.54132498367008399101975596264, 21.31691373702911529378501671502, 22.30305291133803150443303807899

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