↑

Properties

Label 1-967-967.69-r0-0-0
Degree $1$
Conductor $967$
Sign $0.0526 - 0.998i$
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  + (−0.0682 + 0.997i)2-s + (−0.334 + 0.942i)3-s + (−0.990 − 0.136i)4-s + (0.962 + 0.269i)5-s + (−0.917 − 0.398i)6-s + (0.854 − 0.519i)7-s + (0.203 − 0.979i)8-s + (−0.775 − 0.631i)9-s + (−0.334 + 0.942i)10-s + (−0.775 − 0.631i)11-s + (0.460 − 0.887i)12-s + (−0.775 + 0.631i)13-s + (0.460 + 0.887i)14-s + (−0.576 + 0.816i)15-s + (0.962 + 0.269i)16-s + (−0.576 − 0.816i)17-s + ⋯
L(s)  = 1  + (−0.0682 + 0.997i)2-s + (−0.334 + 0.942i)3-s + (−0.990 − 0.136i)4-s + (0.962 + 0.269i)5-s + (−0.917 − 0.398i)6-s + (0.854 − 0.519i)7-s + (0.203 − 0.979i)8-s + (−0.775 − 0.631i)9-s + (−0.334 + 0.942i)10-s + (−0.775 − 0.631i)11-s + (0.460 − 0.887i)12-s + (−0.775 + 0.631i)13-s + (0.460 + 0.887i)14-s + (−0.576 + 0.816i)15-s + (0.962 + 0.269i)16-s + (−0.576 − 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0526 - 0.998i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.0526 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1112850568 + 0.1055680540i\)
\(L(\frac12)\) \(\approx\) \(-0.1112850568 + 0.1055680540i\)
\(L(1)\) \(\approx\) \(0.5373076668 + 0.4877293233i\)
\(L(1)\) \(\approx\) \(0.5373076668 + 0.4877293233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.0682 + 0.997i)T \)
3 \( 1 + (-0.334 + 0.942i)T \)
5 \( 1 + (0.962 + 0.269i)T \)
7 \( 1 + (0.854 - 0.519i)T \)
11 \( 1 + (-0.775 - 0.631i)T \)
13 \( 1 + (-0.775 + 0.631i)T \)
17 \( 1 + (-0.576 - 0.816i)T \)
19 \( 1 + (-0.334 + 0.942i)T \)
23 \( 1 + (-0.990 - 0.136i)T \)
29 \( 1 + (-0.775 + 0.631i)T \)
31 \( 1 + (0.203 - 0.979i)T \)
37 \( 1 + (-0.775 + 0.631i)T \)
41 \( 1 + (-0.917 - 0.398i)T \)
43 \( 1 + (-0.576 + 0.816i)T \)
47 \( 1 + (-0.576 + 0.816i)T \)
53 \( 1 + (-0.990 - 0.136i)T \)
59 \( 1 + (-0.990 + 0.136i)T \)
61 \( 1 + (-0.917 - 0.398i)T \)
67 \( 1 + (0.203 + 0.979i)T \)
71 \( 1 + (-0.0682 - 0.997i)T \)
73 \( 1 + (-0.990 + 0.136i)T \)
79 \( 1 + (-0.990 + 0.136i)T \)
83 \( 1 + (-0.917 - 0.398i)T \)
89 \( 1 + (0.203 + 0.979i)T \)
97 \( 1 + T \)
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

−21.24268165334423408638051211826, −20.203865923731372320645542079308, −19.76657088664939680622608352807, −18.58744161673105153069071608940, −18.10081458723604087702478590039, −17.36022499959452095388183346858, −17.19016253265333622506792782148, −15.436523096992432068207416630136, −14.48795631992809246204742935367, −13.66296731000311473102493861818, −12.943529273308686514214381443566, −12.43321847897511087155178593108, −11.62416556486497235317555204953, −10.66554412271515882552153450811, −10.072051098023081490215295126080, −8.88754107134705957629201028727, −8.26125998078039179072940935891, −7.335905596418963371810273092704, −6.02357099481516514938958053655, −5.19494368705630672126589521803, −4.67412313283647078596480387035, −2.90130035360030212890008798853, −1.98813903549957820535658381447, −1.73125469506348683611846266482, −0.06451921864673651416966872493, 1.73110097732483569506296598378, 3.17030444369045381751212370918, 4.3896548430592997410711035690, 4.98510853934440490801248985703, 5.771730419555679433215527284137, 6.55662742855891749135246984205, 7.62555313451905430049172822038, 8.516223404235554067190313972661, 9.44137268579911870543429894408, 10.102203028134698549253168258958, 10.79065501417287160492354061487, 11.7952914523760585450279578373, 13.12111310769959062782902307202, 14.03784395987301587263932282837, 14.38055525872337421466518888989, 15.22525392296702058947753125015, 16.18226983910399402836957091843, 16.81931125205031536291431212143, 17.36386374082935869789392084686, 18.1728156113361800078348556489, 18.790246800032379776763045896118, 20.27818641653367853728204243866, 20.98486643306178538605446889047, 21.73539928670852522759271673389, 22.28556890089210689617724512082

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