↑

Properties

Label 1-967-967.60-r0-0-0
Degree $1$
Conductor $967$
Sign $0.998 - 0.0615i$
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  + (−0.0422 − 0.999i)2-s + (0.833 − 0.552i)3-s + (−0.996 + 0.0844i)4-s + (0.991 + 0.129i)5-s + (−0.587 − 0.809i)6-s + (−0.759 + 0.651i)7-s + (0.126 + 0.991i)8-s + (0.389 − 0.921i)9-s + (0.0876 − 0.996i)10-s + (−0.750 + 0.660i)11-s + (−0.783 + 0.620i)12-s + (−0.197 + 0.980i)13-s + (0.682 + 0.730i)14-s + (0.898 − 0.439i)15-s + (0.985 − 0.168i)16-s + (−0.371 + 0.928i)17-s + ⋯
L(s)  = 1  + (−0.0422 − 0.999i)2-s + (0.833 − 0.552i)3-s + (−0.996 + 0.0844i)4-s + (0.991 + 0.129i)5-s + (−0.587 − 0.809i)6-s + (−0.759 + 0.651i)7-s + (0.126 + 0.991i)8-s + (0.389 − 0.921i)9-s + (0.0876 − 0.996i)10-s + (−0.750 + 0.660i)11-s + (−0.783 + 0.620i)12-s + (−0.197 + 0.980i)13-s + (0.682 + 0.730i)14-s + (0.898 − 0.439i)15-s + (0.985 − 0.168i)16-s + (−0.371 + 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.998 - 0.0615i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.998 - 0.0615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.532863431 - 0.04718797163i\)
\(L(\frac12)\) \(\approx\) \(1.532863431 - 0.04718797163i\)
\(L(1)\) \(\approx\) \(1.141504667 - 0.4027857081i\)
\(L(1)\) \(\approx\) \(1.141504667 - 0.4027857081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.0422 - 0.999i)T \)
3 \( 1 + (0.833 - 0.552i)T \)
5 \( 1 + (0.991 + 0.129i)T \)
7 \( 1 + (-0.759 + 0.651i)T \)
11 \( 1 + (-0.750 + 0.660i)T \)
13 \( 1 + (-0.197 + 0.980i)T \)
17 \( 1 + (-0.371 + 0.928i)T \)
19 \( 1 + (-0.986 - 0.161i)T \)
23 \( 1 + (0.972 - 0.232i)T \)
29 \( 1 + (-0.984 + 0.174i)T \)
31 \( 1 + (-0.454 + 0.890i)T \)
37 \( 1 + (0.999 - 0.0260i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (0.505 + 0.862i)T \)
47 \( 1 + (0.228 + 0.973i)T \)
53 \( 1 + (-0.158 - 0.987i)T \)
59 \( 1 + (0.997 - 0.0649i)T \)
61 \( 1 + (0.538 - 0.842i)T \)
67 \( 1 + (-0.544 + 0.838i)T \)
71 \( 1 + (-0.844 - 0.536i)T \)
73 \( 1 + (-0.158 + 0.987i)T \)
79 \( 1 + (0.672 + 0.739i)T \)
83 \( 1 + (0.880 + 0.474i)T \)
89 \( 1 + (0.998 + 0.0520i)T \)
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

−21.95774769259678677812738466428, −20.9164664533010374746457246551, −20.37522891412980088369375156698, −19.22274588775495069063197246316, −18.63279739657000068840583385015, −17.629471974469137161154614748278, −16.76764215115056833828102791227, −16.31103682223369333806703565119, −15.35838836450774132447992035987, −14.765047422636709427375818173371, −13.73636710587743466726704948019, −13.27950318561516684988743166571, −12.860752767245834102728621669770, −10.73629201318347343594509107938, −10.26693457630868120031914960128, −9.33431806781895278162415847901, −8.89150601428032891084626853692, −7.77206510446192469033159597521, −7.13331314853515946668720104232, −5.937903050313562775570110818890, −5.30425878281753857470871819698, −4.28968431459118303632390496271, −3.28562888569355774780687403005, −2.37034914618286240029079419975, −0.59188271507008524544438916479, 1.42964216377222884503583668612, 2.29353319380254814049507559491, 2.68027359464073889080622186690, 3.85322244658855168910074709424, 4.92784784677871518333897579705, 6.111742849370300475164355924579, 6.90035038726144797752325083050, 8.150724956665584288381659316424, 9.11545792794496117583281742630, 9.40889139238143125441536069688, 10.321849529036487847123803598697, 11.24859973930394941776027126076, 12.603725236915406614545438175078, 12.8473269738463996707751082007, 13.3888511576333988584233328069, 14.665593576489950883745939508697, 14.8112411866540929877588114201, 16.32971291236509663800142752146, 17.44745357457244195388286452445, 18.024758424727729045786279443222, 18.91261417579433783465737734067, 19.217692633212178791909425985239, 20.154598672860786787429519122069, 21.0562309235878986940565132252, 21.4584212101807996439061990086

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