Properties

Label 1-967-967.45-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

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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: $\chi_{967} (45, \cdot )$
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 \( 1 + (0.623 + 0.781i)T \)
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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

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

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