Properties

Label 1-967-967.43-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.244 - 0.969i$
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  + (0.818 + 0.574i)2-s + (−0.972 + 0.232i)3-s + (0.341 + 0.940i)4-s + (−0.998 − 0.0520i)5-s + (−0.929 − 0.368i)6-s + (0.941 + 0.337i)7-s + (−0.260 + 0.965i)8-s + (0.892 − 0.451i)9-s + (−0.787 − 0.615i)10-s + (−0.608 + 0.793i)11-s + (−0.549 − 0.835i)12-s + (0.383 − 0.923i)13-s + (0.576 + 0.816i)14-s + (0.983 − 0.181i)15-s + (−0.767 + 0.641i)16-s + (0.710 + 0.703i)17-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)2-s + (−0.972 + 0.232i)3-s + (0.341 + 0.940i)4-s + (−0.998 − 0.0520i)5-s + (−0.929 − 0.368i)6-s + (0.941 + 0.337i)7-s + (−0.260 + 0.965i)8-s + (0.892 − 0.451i)9-s + (−0.787 − 0.615i)10-s + (−0.608 + 0.793i)11-s + (−0.549 − 0.835i)12-s + (0.383 − 0.923i)13-s + (0.576 + 0.816i)14-s + (0.983 − 0.181i)15-s + (−0.767 + 0.641i)16-s + (0.710 + 0.703i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.244 - 0.969i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.244 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3586331647 + 0.4602103023i\)
\(L(\frac12)\) \(\approx\) \(-0.3586331647 + 0.4602103023i\)
\(L(1)\) \(\approx\) \(0.7981268211 + 0.5862632593i\)
\(L(1)\) \(\approx\) \(0.7981268211 + 0.5862632593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.818 + 0.574i)T \)
3 \( 1 + (-0.972 + 0.232i)T \)
5 \( 1 + (-0.998 - 0.0520i)T \)
7 \( 1 + (0.941 + 0.337i)T \)
11 \( 1 + (-0.608 + 0.793i)T \)
13 \( 1 + (0.383 - 0.923i)T \)
17 \( 1 + (0.710 + 0.703i)T \)
19 \( 1 + (-0.997 - 0.0649i)T \)
23 \( 1 + (0.750 - 0.660i)T \)
29 \( 1 + (-0.241 + 0.970i)T \)
31 \( 1 + (-0.982 - 0.187i)T \)
37 \( 1 + (0.815 + 0.579i)T \)
41 \( 1 + (-0.682 + 0.730i)T \)
43 \( 1 + (-0.914 - 0.404i)T \)
47 \( 1 + (0.395 + 0.918i)T \)
53 \( 1 + (-0.247 + 0.968i)T \)
59 \( 1 + (0.999 - 0.0260i)T \)
61 \( 1 + (-0.974 - 0.225i)T \)
67 \( 1 + (-0.653 - 0.756i)T \)
71 \( 1 + (0.0876 + 0.996i)T \)
73 \( 1 + (-0.247 - 0.968i)T \)
79 \( 1 + (-0.602 + 0.797i)T \)
83 \( 1 + (-0.677 - 0.735i)T \)
89 \( 1 + (-0.328 + 0.944i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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

−21.263167377580981107426699728344, −20.41996054854027716590184908893, −19.28123090856363837127191143076, −18.80105724948484030437740720475, −18.136610469976483339080613558570, −16.808706489583233479712366287296, −16.27786209529649864008538454139, −15.373015424265419012609106807841, −14.58902296071549601697430557021, −13.623380775992565135326253479361, −12.93889464456638499294664601396, −11.89896468857480552052461067634, −11.395688175182349002181853008861, −10.99120906279730540360443097666, −10.117906631059000898305137155523, −8.72126189224541162056967925189, −7.580906898235583255380523542372, −6.921784065585463484089426346530, −5.80470000151113567793996601733, −5.04148149205328521196206249262, −4.27605587130437298497584071961, −3.48670123260210398637578300272, −2.07877528314369187176411866879, −1.03882678734159077178984769407, −0.118922339166801557147543467951, 1.46819543861219364066712807422, 2.95386430040511306352582711014, 4.03460118688032530808550103449, 4.76974874117726937361411818560, 5.34434878248745814095661947378, 6.30696436358043007789722470735, 7.33137770254721718363474843416, 7.95381708048042194786020862232, 8.7850702539526668743952227594, 10.473182723683963618547305255666, 10.99222576698690979171305058761, 11.81603826775488596026480209319, 12.69640654329102571748628371147, 12.88567908406225989244978211117, 14.61857745187271622622896840040, 15.101354646015937372716098656937, 15.53815584549250679811714818975, 16.604988851983448983237284802220, 17.084918972704407784976677744305, 18.09077550696451908000687486820, 18.622868303945854693173541492623, 20.15064070695397176822489728305, 20.72937531314624795684904133214, 21.504428867431772611730216354831, 22.26623876003757006200043947776

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