Properties

Label 1-967-967.546-r1-0-0
Degree $1$
Conductor $967$
Sign $0.109 - 0.993i$
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  + (−0.648 + 0.761i)2-s + (0.990 − 0.136i)3-s + (−0.158 − 0.987i)4-s + (−0.746 − 0.665i)5-s + (−0.538 + 0.842i)6-s + (−0.113 + 0.993i)7-s + (0.854 + 0.519i)8-s + (0.962 − 0.269i)9-s + (0.990 − 0.136i)10-s + (0.962 − 0.269i)11-s + (−0.291 − 0.956i)12-s + (0.715 − 0.699i)13-s + (−0.682 − 0.730i)14-s + (−0.829 − 0.557i)15-s + (−0.949 + 0.313i)16-s + (−0.0682 − 0.997i)17-s + ⋯
L(s)  = 1  + (−0.648 + 0.761i)2-s + (0.990 − 0.136i)3-s + (−0.158 − 0.987i)4-s + (−0.746 − 0.665i)5-s + (−0.538 + 0.842i)6-s + (−0.113 + 0.993i)7-s + (0.854 + 0.519i)8-s + (0.962 − 0.269i)9-s + (0.990 − 0.136i)10-s + (0.962 − 0.269i)11-s + (−0.291 − 0.956i)12-s + (0.715 − 0.699i)13-s + (−0.682 − 0.730i)14-s + (−0.829 − 0.557i)15-s + (−0.949 + 0.313i)16-s + (−0.0682 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.109 - 0.993i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (546, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.109 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.149178360 - 1.029623242i\)
\(L(\frac12)\) \(\approx\) \(1.149178360 - 1.029623242i\)
\(L(1)\) \(\approx\) \(0.9894373203 + 0.005986561268i\)
\(L(1)\) \(\approx\) \(0.9894373203 + 0.005986561268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.648 + 0.761i)T \)
3 \( 1 + (0.990 - 0.136i)T \)
5 \( 1 + (-0.746 - 0.665i)T \)
7 \( 1 + (-0.113 + 0.993i)T \)
11 \( 1 + (0.962 - 0.269i)T \)
13 \( 1 + (0.715 - 0.699i)T \)
17 \( 1 + (-0.0682 - 0.997i)T \)
19 \( 1 + (-0.613 - 0.789i)T \)
23 \( 1 + (0.775 - 0.631i)T \)
29 \( 1 + (-0.962 - 0.269i)T \)
31 \( 1 + (-0.877 + 0.480i)T \)
37 \( 1 + (0.715 - 0.699i)T \)
41 \( 1 + (-0.460 - 0.887i)T \)
43 \( 1 + (-0.898 + 0.439i)T \)
47 \( 1 + (0.829 + 0.557i)T \)
53 \( 1 + (0.934 + 0.356i)T \)
59 \( 1 + (0.934 - 0.356i)T \)
61 \( 1 + (-0.998 - 0.0455i)T \)
67 \( 1 + (-0.854 + 0.519i)T \)
71 \( 1 + (-0.334 + 0.942i)T \)
73 \( 1 + (0.934 - 0.356i)T \)
79 \( 1 + (-0.934 + 0.356i)T \)
83 \( 1 + (0.538 - 0.842i)T \)
89 \( 1 + (-0.0227 - 0.999i)T \)
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

−21.54203833464687773569698213090, −20.69162565820750659604113474002, −19.98034528410817893435033316277, −19.49836694492767154713144926203, −18.84387034807367850784293544335, −18.18158296959121925234010468947, −16.77775436696931134347522887454, −16.609236821020622419259508393053, −15.20400153137664928946375578370, −14.66105286049681377396455581834, −13.63870219812415423425512773122, −13.02236958671276181026702199239, −11.94465498829372113006681999578, −11.070194358209008240079850122, −10.45250342593796243529108864025, −9.60312668985881545964817986677, −8.76260948387781730555402362711, −7.99235146762286802188102761062, −7.215701849397240489434887142084, −6.55625001367781523444789659861, −4.32305304846962743577511908294, −3.81086027005353602567335071262, −3.36398160510718016156108033641, −1.96250740273876652198482256151, −1.23681213670670281951653073618, 0.38227050013308018913781184485, 1.35641655470872792343211804485, 2.56068697942231249218861179985, 3.727551855697064374120494033239, 4.74493565217788703472947925499, 5.7254744277551178888101393200, 6.814365879655665825429602453763, 7.55350586378982372715179519251, 8.57597959488512404344365338995, 8.89459522756806004084393493494, 9.43063159524736657097870869599, 10.799719566012518175771228870009, 11.69236872024620314482686039129, 12.78769275299020127517140059212, 13.449206904319165399716361672714, 14.60766623379144807628108119324, 15.089500097004271698986654406754, 15.8219497977839810913667106349, 16.3981157477132622466663366785, 17.43658429563379936770461935635, 18.50775025354360361691099747470, 18.84978242066629935912889525488, 19.79633815409570773636754259032, 20.195886257718211548224875681962, 21.1468424942127799046640430864

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