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Properties

Label 1-967-967.457-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.340 - 0.940i$
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  + (0.00975 − 0.999i)2-s + (0.477 − 0.878i)3-s + (−0.999 − 0.0195i)4-s + (−0.592 − 0.805i)5-s + (−0.874 − 0.485i)6-s + (−0.996 + 0.0779i)7-s + (−0.0292 + 0.999i)8-s + (−0.544 − 0.838i)9-s + (−0.811 + 0.584i)10-s + (−0.696 + 0.717i)11-s + (−0.494 + 0.869i)12-s + (0.696 + 0.717i)13-s + (0.0682 + 0.997i)14-s + (−0.990 + 0.136i)15-s + (0.999 + 0.0390i)16-s + (0.0876 + 0.996i)17-s + ⋯
L(s)  = 1  + (0.00975 − 0.999i)2-s + (0.477 − 0.878i)3-s + (−0.999 − 0.0195i)4-s + (−0.592 − 0.805i)5-s + (−0.874 − 0.485i)6-s + (−0.996 + 0.0779i)7-s + (−0.0292 + 0.999i)8-s + (−0.544 − 0.838i)9-s + (−0.811 + 0.584i)10-s + (−0.696 + 0.717i)11-s + (−0.494 + 0.869i)12-s + (0.696 + 0.717i)13-s + (0.0682 + 0.997i)14-s + (−0.990 + 0.136i)15-s + (0.999 + 0.0390i)16-s + (0.0876 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.340 - 0.940i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.340 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4946211771 - 0.7055449875i\)
\(L(\frac12)\) \(\approx\) \(0.4946211771 - 0.7055449875i\)
\(L(1)\) \(\approx\) \(0.5152685323 - 0.5432445863i\)
\(L(1)\) \(\approx\) \(0.5152685323 - 0.5432445863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.00975 - 0.999i)T \)
3 \( 1 + (0.477 - 0.878i)T \)
5 \( 1 + (-0.592 - 0.805i)T \)
7 \( 1 + (-0.996 + 0.0779i)T \)
11 \( 1 + (-0.696 + 0.717i)T \)
13 \( 1 + (0.696 + 0.717i)T \)
17 \( 1 + (0.0876 + 0.996i)T \)
19 \( 1 + (-0.389 - 0.921i)T \)
23 \( 1 + (-0.909 + 0.416i)T \)
29 \( 1 + (-0.316 - 0.948i)T \)
31 \( 1 + (-0.967 - 0.250i)T \)
37 \( 1 + (-0.126 + 0.991i)T \)
41 \( 1 + (0.576 + 0.816i)T \)
43 \( 1 + (-0.353 - 0.935i)T \)
47 \( 1 + (0.511 - 0.859i)T \)
53 \( 1 + (0.203 + 0.979i)T \)
59 \( 1 + (0.892 - 0.451i)T \)
61 \( 1 + (-0.576 - 0.816i)T \)
67 \( 1 + (0.967 - 0.250i)T \)
71 \( 1 + (0.00975 + 0.999i)T \)
73 \( 1 + (0.203 - 0.979i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (-0.668 + 0.744i)T \)
89 \( 1 + (0.967 - 0.250i)T \)
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

−22.07496650974552941157866345502, −21.16440731011310723592195559901, −20.16026581590933201606901829849, −19.31336753037113693974999474797, −18.57405165133039059820173910205, −17.97349855059910292392284489295, −16.492707601077974109953185357, −16.09970803085641189066392998485, −15.72003952410685580456891568786, −14.70405467567278525129600995573, −14.161318885204243624716093776626, −13.322584942791211476063966235239, −12.423238611045074457055684987316, −10.93938845900961038088701971463, −10.39906674258085324060580119184, −9.555541591400075784845501225803, −8.61701417834043373819759003777, −7.912734228783411872158864557750, −7.11366854971710300975613460822, −5.99982565649433720091719065938, −5.39778951914192197579221942137, −4.01340460747972504236977573582, −3.51807449552897424728557255157, −2.70837702352091631377906811078, −0.363982367725768677484760006091, 0.46842586564098433338306592864, 1.641386998385786290015400079564, 2.40737295300652890237692157934, 3.59891642547695325868611025412, 4.15092597089181666901020392552, 5.44875278943403906979381942240, 6.470881666076883545542072765, 7.63348405994089208381030402586, 8.39383496783864838736279407320, 9.135429132480973137749154868774, 9.83298593577032479758389195449, 11.0177967319775217119701822415, 11.95465513411656593519383644708, 12.49669403571154373174214748286, 13.27508714362781742647076178991, 13.53292594605635534005471758180, 14.917468806687460804203748203913, 15.63925242739776094632243296603, 16.817182951256509705691969790032, 17.55298531727503423938963114621, 18.6410093160101860490150238797, 18.97487791277229475655167821330, 19.97927029344526480062361228220, 20.11850441571075965219407835452, 21.11753115047460951138808156700

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