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Properties

Label 1-967-967.19-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: $\chi_{967} (19, \cdot )$
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

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

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