Properties

Label 1-967-967.509-r1-0-0
Degree $1$
Conductor $967$
Sign $0.992 - 0.125i$
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  + (0.100 − 0.994i)2-s + (−0.909 − 0.416i)3-s + (−0.979 − 0.200i)4-s + (−0.701 + 0.712i)5-s + (−0.505 + 0.862i)6-s + (0.971 + 0.238i)7-s + (−0.297 + 0.954i)8-s + (0.653 + 0.756i)9-s + (0.638 + 0.769i)10-s + (0.592 − 0.805i)11-s + (0.807 + 0.589i)12-s + (0.993 − 0.110i)13-s + (0.334 − 0.942i)14-s + (0.934 − 0.356i)15-s + (0.919 + 0.392i)16-s + (0.787 + 0.615i)17-s + ⋯
L(s)  = 1  + (0.100 − 0.994i)2-s + (−0.909 − 0.416i)3-s + (−0.979 − 0.200i)4-s + (−0.701 + 0.712i)5-s + (−0.505 + 0.862i)6-s + (0.971 + 0.238i)7-s + (−0.297 + 0.954i)8-s + (0.653 + 0.756i)9-s + (0.638 + 0.769i)10-s + (0.592 − 0.805i)11-s + (0.807 + 0.589i)12-s + (0.993 − 0.110i)13-s + (0.334 − 0.942i)14-s + (0.934 − 0.356i)15-s + (0.919 + 0.392i)16-s + (0.787 + 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.688536903 - 0.1063489737i\)
\(L(\frac12)\) \(\approx\) \(1.688536903 - 0.1063489737i\)
\(L(1)\) \(\approx\) \(0.8779988300 - 0.3087851940i\)
\(L(1)\) \(\approx\) \(0.8779988300 - 0.3087851940i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.100 - 0.994i)T \)
3 \( 1 + (-0.909 - 0.416i)T \)
5 \( 1 + (-0.701 + 0.712i)T \)
7 \( 1 + (0.971 + 0.238i)T \)
11 \( 1 + (0.592 - 0.805i)T \)
13 \( 1 + (0.993 - 0.110i)T \)
17 \( 1 + (0.787 + 0.615i)T \)
19 \( 1 + (0.837 + 0.547i)T \)
23 \( 1 + (0.696 + 0.717i)T \)
29 \( 1 + (0.184 + 0.982i)T \)
31 \( 1 + (0.867 - 0.497i)T \)
37 \( 1 + (0.705 + 0.708i)T \)
41 \( 1 + (0.0682 + 0.997i)T \)
43 \( 1 + (-0.997 + 0.0649i)T \)
47 \( 1 + (-0.304 + 0.952i)T \)
53 \( 1 + (-0.877 - 0.480i)T \)
59 \( 1 + (-0.922 - 0.386i)T \)
61 \( 1 + (0.898 + 0.439i)T \)
67 \( 1 + (0.864 - 0.502i)T \)
71 \( 1 + (0.811 - 0.584i)T \)
73 \( 1 + (-0.877 + 0.480i)T \)
79 \( 1 + (-0.998 + 0.0520i)T \)
83 \( 1 + (0.728 + 0.684i)T \)
89 \( 1 + (0.00325 + 0.999i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−21.659130547541280172711888886495, −20.8856060873521570658440383653, −20.26052756914684039852420961846, −18.8843177403754165198402233084, −18.12550240162167020283131861656, −17.354042823518890927687101672291, −16.84559641278369362568285137803, −16.00555566124737562665186115369, −15.48173430274047500101899849525, −14.65807474847242025097371092754, −13.746889823091865954794002547420, −12.7230936314741210944281675568, −11.89780071229633825115714411605, −11.34782401539450647248376335858, −10.12873801789905586841605898097, −9.22864535610077319482209477602, −8.439433752516837851414080868294, −7.48495741597395848899360343656, −6.809571626204623053446557796002, −5.68396413314557545040550968372, −4.86047923833178897241294888982, −4.42380998051421290003008716223, −3.53395592242808373236565504124, −1.23081269750507853896314558474, −0.56901370192653121223292848007, 1.03536132119343541492147974370, 1.43753379486155257056512640496, 2.98996127191107461713315800753, 3.75505488378646487024886406459, 4.79540523775624963345949688960, 5.6949524831233167726514540909, 6.50716686155029302884580582382, 7.92224120405545661717737220044, 8.25713999571025712604557187628, 9.662529085394271196163351146600, 10.679990823351081200444620301760, 11.29404001792469920431020996551, 11.61290153041558391368745473857, 12.425597505305673116702969628204, 13.45789565509189062593092149521, 14.21794385448344821242913627269, 15.00219355555280496863176044173, 16.116625451802695971760101718076, 17.055917392467823196246872472516, 17.903085514309804119819261940027, 18.58928068166662593527363394787, 18.94719141954463749377011198556, 19.86440795346652468926999703102, 20.893241728617552242698202032344, 21.650491204173363086639446380183

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