↑

Properties

Label 1-967-967.421-r0-0-0
Degree $1$
Conductor $967$
Sign $0.103 + 0.994i$
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  + (−0.648 + 0.761i)2-s + (−0.990 + 0.136i)3-s + (−0.158 − 0.987i)4-s + (0.746 + 0.665i)5-s + (0.538 − 0.842i)6-s + (0.113 − 0.993i)7-s + (0.854 + 0.519i)8-s + (0.962 − 0.269i)9-s + (−0.990 + 0.136i)10-s + (0.962 − 0.269i)11-s + (0.291 + 0.956i)12-s + (−0.715 + 0.699i)13-s + (0.682 + 0.730i)14-s + (−0.829 − 0.557i)15-s + (−0.949 + 0.313i)16-s + (−0.0682 − 0.997i)17-s + ⋯
L(s)  = 1  + (−0.648 + 0.761i)2-s + (−0.990 + 0.136i)3-s + (−0.158 − 0.987i)4-s + (0.746 + 0.665i)5-s + (0.538 − 0.842i)6-s + (0.113 − 0.993i)7-s + (0.854 + 0.519i)8-s + (0.962 − 0.269i)9-s + (−0.990 + 0.136i)10-s + (0.962 − 0.269i)11-s + (0.291 + 0.956i)12-s + (−0.715 + 0.699i)13-s + (0.682 + 0.730i)14-s + (−0.829 − 0.557i)15-s + (−0.949 + 0.313i)16-s + (−0.0682 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.103 + 0.994i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.103 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6357347969 + 0.5731403173i\)
\(L(\frac12)\) \(\approx\) \(0.6357347969 + 0.5731403173i\)
\(L(1)\) \(\approx\) \(0.6403761527 + 0.2799458475i\)
\(L(1)\) \(\approx\) \(0.6403761527 + 0.2799458475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.648 + 0.761i)T \)
3 \( 1 + (-0.990 + 0.136i)T \)
5 \( 1 + (0.746 + 0.665i)T \)
7 \( 1 + (0.113 - 0.993i)T \)
11 \( 1 + (0.962 - 0.269i)T \)
13 \( 1 + (-0.715 + 0.699i)T \)
17 \( 1 + (-0.0682 - 0.997i)T \)
19 \( 1 + (0.613 + 0.789i)T \)
23 \( 1 + (-0.775 + 0.631i)T \)
29 \( 1 + (0.962 + 0.269i)T \)
31 \( 1 + (-0.877 + 0.480i)T \)
37 \( 1 + (-0.715 + 0.699i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (0.898 - 0.439i)T \)
47 \( 1 + (-0.829 - 0.557i)T \)
53 \( 1 + (0.934 + 0.356i)T \)
59 \( 1 + (0.934 - 0.356i)T \)
61 \( 1 + (-0.998 - 0.0455i)T \)
67 \( 1 + (0.854 - 0.519i)T \)
71 \( 1 + (-0.334 + 0.942i)T \)
73 \( 1 + (0.934 - 0.356i)T \)
79 \( 1 + (0.934 - 0.356i)T \)
83 \( 1 + (0.538 - 0.842i)T \)
89 \( 1 + (0.0227 + 0.999i)T \)
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

−21.57235830709056310728258903769, −20.92681328765278270549699579272, −19.8165522244747426319510824528, −19.307785131437669545707098958659, −18.053156557990666095872251371064, −17.7927368956389981396622796631, −17.14182353362555371934314753345, −16.36956919189914087479523284749, −15.52975632281593344057309244530, −14.28736563854353165155553770120, −13.10947587604552168127343082445, −12.39227512995060073929540415317, −12.12132552347147786505804126753, −11.11214930176926450568572878874, −10.21917643853732780291405787569, −9.51179354576516454544786340183, −8.80125552626577953564848824484, −7.81624575420360057381606652249, −6.6799627539652269245070965005, −5.77874926359139217228612096380, −4.929089555409152138325286673902, −4.00830357608357195022715303891, −2.45423410247985368140631380406, −1.75722439071103725229414021007, −0.67244476082198567800666989040, 0.99365326033217144355557856902, 1.85745325367058561992789822106, 3.64534584753212088996717319160, 4.74366455094134288618068484355, 5.53926665712314528708966295778, 6.53788797198039114456247246913, 6.92805075495847598003086093899, 7.71473417378797293531646060027, 9.240421025764895710591487758725, 9.80530119958804590120361473150, 10.441482804915852808717241186930, 11.29248935370543422279280685762, 12.034825365424301317727281080174, 13.57286939213270870116370932062, 14.1019125896321192293678543972, 14.74029584430137344615262559687, 16.0163753858459379108290672512, 16.5421951188411023692086473979, 17.21466216083800824471679512356, 17.8230883758574324212634401300, 18.44164437955625807648630518829, 19.3622342518275819257588010629, 20.19978963024623851815908807678, 21.34622717479961333068039190724, 22.17568380198685632079380901716

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