↑

Properties

Label 1-967-967.191-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.974 - 0.225i$
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  + (0.981 − 0.193i)2-s + (0.864 − 0.502i)3-s + (0.924 − 0.380i)4-s + (−0.993 − 0.116i)5-s + (0.750 − 0.660i)6-s + (−0.00975 − 0.999i)7-s + (0.833 − 0.552i)8-s + (0.494 − 0.869i)9-s + (−0.996 + 0.0779i)10-s + (−0.957 − 0.288i)11-s + (0.608 − 0.793i)12-s + (0.957 − 0.288i)13-s + (−0.203 − 0.979i)14-s + (−0.917 + 0.398i)15-s + (0.710 − 0.703i)16-s + (−0.184 + 0.982i)17-s + ⋯
L(s)  = 1  + (0.981 − 0.193i)2-s + (0.864 − 0.502i)3-s + (0.924 − 0.380i)4-s + (−0.993 − 0.116i)5-s + (0.750 − 0.660i)6-s + (−0.00975 − 0.999i)7-s + (0.833 − 0.552i)8-s + (0.494 − 0.869i)9-s + (−0.996 + 0.0779i)10-s + (−0.957 − 0.288i)11-s + (0.608 − 0.793i)12-s + (0.957 − 0.288i)13-s + (−0.203 − 0.979i)14-s + (−0.917 + 0.398i)15-s + (0.710 − 0.703i)16-s + (−0.184 + 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.974 - 0.225i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.974 - 0.225i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4800240799 - 4.194765137i\)
\(L(\frac12)\) \(\approx\) \(0.4800240799 - 4.194765137i\)
\(L(1)\) \(\approx\) \(1.723571477 - 1.294989529i\)
\(L(1)\) \(\approx\) \(1.723571477 - 1.294989529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.981 - 0.193i)T \)
3 \( 1 + (0.864 - 0.502i)T \)
5 \( 1 + (-0.993 - 0.116i)T \)
7 \( 1 + (-0.00975 - 0.999i)T \)
11 \( 1 + (-0.957 - 0.288i)T \)
13 \( 1 + (0.957 - 0.288i)T \)
17 \( 1 + (-0.184 + 0.982i)T \)
19 \( 1 + (0.145 - 0.989i)T \)
23 \( 1 + (0.668 - 0.744i)T \)
29 \( 1 + (-0.987 - 0.155i)T \)
31 \( 1 + (0.353 + 0.935i)T \)
37 \( 1 + (0.822 + 0.568i)T \)
41 \( 1 + (-0.962 + 0.269i)T \)
43 \( 1 + (-0.592 - 0.805i)T \)
47 \( 1 + (0.260 - 0.965i)T \)
53 \( 1 + (-0.576 - 0.816i)T \)
59 \( 1 + (-0.998 + 0.0585i)T \)
61 \( 1 + (0.962 - 0.269i)T \)
67 \( 1 + (-0.353 + 0.935i)T \)
71 \( 1 + (0.981 + 0.193i)T \)
73 \( 1 + (-0.576 + 0.816i)T \)
79 \( 1 + (-0.874 + 0.485i)T \)
83 \( 1 + (-0.477 - 0.878i)T \)
89 \( 1 + (-0.353 + 0.935i)T \)
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

−21.94300561126761120553352645809, −20.911032493275491953542402442, −20.73283827005292374783435366177, −19.809402297148881036329682502853, −18.8173122736239482503264380454, −18.37695527650134873046082787393, −16.64664490064126566911380887273, −15.96576088454794836300618312352, −15.438165901208564939007679152684, −14.97844087789163337033936044977, −14.06237264413544446020181236733, −13.20555372212785356336298432404, −12.51088382291060660342627389155, −11.45990194100689258667644360320, −10.986530156049543916421438315107, −9.72559286555418700122441967653, −8.72202347439648157685371287029, −7.861640454600049760353267148429, −7.371094027891787333819318643, −6.03819782722306771845264297027, −5.09874669669522208370383015890, −4.326650637014457068815024043867, −3.363060302591865344789839077004, −2.809061263753833599837130387325, −1.73808337569615931803071948655, 0.51481954853230322373623631327, 1.45296219775822510187283498687, 2.78648803354948611595027635008, 3.494421251654302457598956614852, 4.16036254358882637399114250859, 5.14293145271694950907513237026, 6.53328103982539736986808882342, 7.10768328089588079664942651066, 8.02145686999165250484258116782, 8.610341063695575254278432294431, 10.15181386930226380552090307282, 10.87998834298384091147841764831, 11.60050515394751875794264990814, 12.8814061049202908616764669743, 13.06464611160194114674888427296, 13.83743056671122828751725842238, 14.83967387788298832556671969538, 15.403052392391735085278490531524, 16.09125009339279456618352361585, 17.086258525354840370545962120985, 18.44011468911219123906032940401, 19.05775463371899318142137289615, 19.9315320783307604947047462812, 20.338162838113361372544155903070, 20.95877186584419733064431080403

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