Properties

Label 1-967-967.382-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.837 - 0.546i$
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  + (0.113 − 0.993i)2-s + (−0.460 − 0.887i)3-s + (−0.974 − 0.225i)4-s + (0.829 − 0.557i)5-s + (−0.934 + 0.356i)6-s + (−0.377 − 0.926i)7-s + (−0.334 + 0.942i)8-s + (−0.576 + 0.816i)9-s + (−0.460 − 0.887i)10-s + (−0.576 + 0.816i)11-s + (0.247 + 0.968i)12-s + (0.419 − 0.907i)13-s + (−0.962 + 0.269i)14-s + (−0.877 − 0.480i)15-s + (0.898 + 0.439i)16-s + (0.854 + 0.519i)17-s + ⋯
L(s)  = 1  + (0.113 − 0.993i)2-s + (−0.460 − 0.887i)3-s + (−0.974 − 0.225i)4-s + (0.829 − 0.557i)5-s + (−0.934 + 0.356i)6-s + (−0.377 − 0.926i)7-s + (−0.334 + 0.942i)8-s + (−0.576 + 0.816i)9-s + (−0.460 − 0.887i)10-s + (−0.576 + 0.816i)11-s + (0.247 + 0.968i)12-s + (0.419 − 0.907i)13-s + (−0.962 + 0.269i)14-s + (−0.877 − 0.480i)15-s + (0.898 + 0.439i)16-s + (0.854 + 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.837 - 0.546i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (382, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.837 - 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5712765622 - 1.919856410i\)
\(L(\frac12)\) \(\approx\) \(0.5712765622 - 1.919856410i\)
\(L(1)\) \(\approx\) \(0.6638940483 - 0.8291791817i\)
\(L(1)\) \(\approx\) \(0.6638940483 - 0.8291791817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.113 - 0.993i)T \)
3 \( 1 + (-0.460 - 0.887i)T \)
5 \( 1 + (0.829 - 0.557i)T \)
7 \( 1 + (-0.377 - 0.926i)T \)
11 \( 1 + (-0.576 + 0.816i)T \)
13 \( 1 + (0.419 - 0.907i)T \)
17 \( 1 + (0.854 + 0.519i)T \)
19 \( 1 + (0.998 + 0.0455i)T \)
23 \( 1 + (-0.682 + 0.730i)T \)
29 \( 1 + (0.576 + 0.816i)T \)
31 \( 1 + (0.983 - 0.181i)T \)
37 \( 1 + (0.419 - 0.907i)T \)
41 \( 1 + (0.775 + 0.631i)T \)
43 \( 1 + (-0.0227 - 0.999i)T \)
47 \( 1 + (0.877 + 0.480i)T \)
53 \( 1 + (0.291 + 0.956i)T \)
59 \( 1 + (0.291 - 0.956i)T \)
61 \( 1 + (-0.158 + 0.987i)T \)
67 \( 1 + (0.334 + 0.942i)T \)
71 \( 1 + (-0.917 - 0.398i)T \)
73 \( 1 + (0.291 - 0.956i)T \)
79 \( 1 + (-0.291 + 0.956i)T \)
83 \( 1 + (0.934 - 0.356i)T \)
89 \( 1 + (0.648 - 0.761i)T \)
97 \( 1 + T \)
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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

−21.93275110758694102613262859110, −21.38645557976615664716404059396, −20.815195730630277039632759445340, −19.099276422929421111441394121328, −18.451852600101636704187210290596, −17.91696949341305505340554305003, −16.92168773520132461066750640901, −16.14966220745175025151666266240, −15.80997816746618132664371073638, −14.81434710701624713957931898441, −14.06779841823164648395625723157, −13.49771516688180178337329725351, −12.2169561478723479461422619748, −11.4798801426574432712038381634, −10.23410189884945373557100537431, −9.6803005384732534317976607717, −8.955301199705088281048148464155, −8.06538304304469709029514446885, −6.693416244131174412573884283132, −6.04097946928654226210541770568, −5.53911054137732714757657274660, −4.64814572576819040490529352228, −3.4222570299057806503479499078, −2.672959788014363273132161059275, −0.71799051384183573108204805959, 0.75038546831530640767379620336, 1.22271530159947893919511549770, 2.29838712267092402216156767608, 3.29307960016092513873332444277, 4.53464628542692658381090610172, 5.47239796144829911111569430593, 6.03059958033959850305738564512, 7.437325829182534278179930615688, 8.07443399006640017601239174294, 9.28468930974249562446293446225, 10.276906809790982400809239700718, 10.49690927904189957764975486538, 11.80337793308223421970226585739, 12.55544167058746356622841699489, 13.05638597913548006536790075278, 13.72485549250012727179048974683, 14.34528497215434270704000976057, 15.869582490627566075454871032624, 16.85317116383916710163562748440, 17.743093449566715380483559210734, 17.8652258763485618160098587787, 18.909613101693161013988059399603, 19.90325519329809518011721377849, 20.28380778516916713264152227618, 21.09035161363770463549181578130

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