↑

Properties

Label 1-967-967.99-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.994 - 0.101i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.407 + 0.913i)2-s + (0.560 + 0.828i)3-s + (−0.668 − 0.744i)4-s + (−0.844 + 0.536i)5-s + (−0.984 + 0.174i)6-s + (−0.977 + 0.212i)7-s + (0.951 − 0.307i)8-s + (−0.371 + 0.928i)9-s + (−0.145 − 0.989i)10-s + (0.987 + 0.155i)11-s + (0.241 − 0.970i)12-s + (0.987 − 0.155i)13-s + (0.203 − 0.979i)14-s + (−0.917 − 0.398i)15-s + (−0.107 + 0.994i)16-s + (0.592 + 0.805i)17-s + ⋯
L(s)  = 1  + (−0.407 + 0.913i)2-s + (0.560 + 0.828i)3-s + (−0.668 − 0.744i)4-s + (−0.844 + 0.536i)5-s + (−0.984 + 0.174i)6-s + (−0.977 + 0.212i)7-s + (0.951 − 0.307i)8-s + (−0.371 + 0.928i)9-s + (−0.145 − 0.989i)10-s + (0.987 + 0.155i)11-s + (0.241 − 0.970i)12-s + (0.987 − 0.155i)13-s + (0.203 − 0.979i)14-s + (−0.917 − 0.398i)15-s + (−0.107 + 0.994i)16-s + (0.592 + 0.805i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.994 - 0.101i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.994 - 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05634566898 + 1.106414984i\)
\(L(\frac12)\) \(\approx\) \(-0.05634566898 + 1.106414984i\)
\(L(1)\) \(\approx\) \(0.5387887446 + 0.7017465166i\)
\(L(1)\) \(\approx\) \(0.5387887446 + 0.7017465166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.407 + 0.913i)T \)
3 \( 1 + (0.560 + 0.828i)T \)
5 \( 1 + (-0.844 + 0.536i)T \)
7 \( 1 + (-0.977 + 0.212i)T \)
11 \( 1 + (0.987 + 0.155i)T \)
13 \( 1 + (0.987 - 0.155i)T \)
17 \( 1 + (0.592 + 0.805i)T \)
19 \( 1 + (0.996 + 0.0779i)T \)
23 \( 1 + (0.924 + 0.380i)T \)
29 \( 1 + (-0.957 - 0.288i)T \)
31 \( 1 + (0.0876 + 0.996i)T \)
37 \( 1 + (0.737 + 0.675i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (-0.184 + 0.982i)T \)
47 \( 1 + (-0.883 + 0.468i)T \)
53 \( 1 + (-0.576 + 0.816i)T \)
59 \( 1 + (0.279 - 0.960i)T \)
61 \( 1 + (0.962 + 0.269i)T \)
67 \( 1 + (0.0876 - 0.996i)T \)
71 \( 1 + (-0.407 - 0.913i)T \)
73 \( 1 + (-0.576 - 0.816i)T \)
79 \( 1 + (0.165 + 0.986i)T \)
83 \( 1 + (0.0487 - 0.998i)T \)
89 \( 1 + (0.0876 - 0.996i)T \)
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

−20.79179188298647683471446566876, −20.45030970660990710138520762295, −19.73290530485800482997084562066, −18.99982562038642407723670777027, −18.687057718591718594174573623698, −17.62847265722665078239526978818, −16.58082739135157229022288484471, −16.16210733066528429588210363165, −14.81309792588469164429520036843, −13.836868743939277277960634382846, −13.134952282240240819664319104712, −12.562346971095694669815129563344, −11.65488866484122530039881220713, −11.241384090189360586147446868702, −9.70150444312890262981844865556, −9.15888645959438777764294019173, −8.49576539605423089828270188059, −7.488318364467019092587204429857, −6.90760831636815051606611887001, −5.55661257006864077210246572128, −3.99786057780301057503602854056, −3.566854968541553262839113204962, −2.68648810540182072155220660860, −1.264242273358246268263830355886, −0.656302128933533326890607303142, 1.29379306465890828858512885517, 3.16541625544235166258193771409, 3.64152090724047687186847171108, 4.602372148569757056365260377921, 5.81128171678071120647726930367, 6.55720231375585848139390684066, 7.55746884704202245341942114813, 8.27924083767066171313136832710, 9.20586756252644297004662087803, 9.70905304864170470613895608152, 10.68435912932016401370743762487, 11.446123636211995587555171075249, 12.790340037062642871880609725226, 13.71018866906353795887462942083, 14.640716724851297220518724591037, 15.04588103558436488308974950714, 15.96888320323989830440902626043, 16.26567497717686205742188905250, 17.18288078749383022892760640759, 18.33459650817486526846132931564, 19.141426520828548387616245026751, 19.55304155565982299265193312053, 20.3206495206321603438525876206, 21.55460581301499360471395722465, 22.46158401408863402099394391440

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