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Properties

Label 1-67-67.5-r1-0-0
Degree $1$
Conductor $67$
Sign $-0.995 + 0.0909i$
Analytic cond. $7.20014$
Root an. cond. $7.20014$
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  + (0.142 + 0.989i)2-s + (0.654 − 0.755i)3-s + (−0.959 + 0.281i)4-s + (−0.841 + 0.540i)5-s + (0.841 + 0.540i)6-s + (0.142 + 0.989i)7-s + (−0.415 − 0.909i)8-s + (−0.142 − 0.989i)9-s + (−0.654 − 0.755i)10-s + (−0.841 + 0.540i)11-s + (−0.415 + 0.909i)12-s + (−0.415 + 0.909i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + (0.654 − 0.755i)3-s + (−0.959 + 0.281i)4-s + (−0.841 + 0.540i)5-s + (0.841 + 0.540i)6-s + (0.142 + 0.989i)7-s + (−0.415 − 0.909i)8-s + (−0.142 − 0.989i)9-s + (−0.654 − 0.755i)10-s + (−0.841 + 0.540i)11-s + (−0.415 + 0.909i)12-s + (−0.415 + 0.909i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.995 + 0.0909i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ -0.995 + 0.0909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04105966147 + 0.9011264993i\)
\(L(\frac12)\) \(\approx\) \(0.04105966147 + 0.9011264993i\)
\(L(1)\) \(\approx\) \(0.7228887601 + 0.5311405182i\)
\(L(1)\) \(\approx\) \(0.7228887601 + 0.5311405182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.142 + 0.989i)T \)
3 \( 1 + (0.654 - 0.755i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.654 + 0.755i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
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

−31.12428229508254945623296187176, −30.39382361410754602531689801452, −28.96642965759389124888629228482, −27.85213929555197594129303452901, −26.93596909586555472111971953484, −26.326158088599581873841314179305, −24.32912202441113712571157199657, −23.279576424686992878176868990326, −22.051932544009657925697098206891, −20.90682622952531770294808696794, −19.97891250178108021601774582625, −19.54263927529894638243519116525, −17.77383748178419873264088090773, −16.26986758504604450494154582642, −15.08198545924106709001245918738, −13.73200220957127501545154565720, −12.76737209243874896248886896853, −11.05567768776117033984998875282, −10.358619843686321236202405867506, −8.81792559617546433036403137115, −7.86518550713319124283000735114, −4.96598847978050130841676262207, −4.07489074185668746361796245798, −2.75330441720830896113377568893, −0.40258129269001914266148884874, 2.534480695464394280759526380529, 4.26912926615679304573442781760, 6.12267950576253272385703684725, 7.37361332389477904773832306684, 8.20104820424699636009203065194, 9.474910471469517320060850339191, 11.78615499884575484360810550795, 12.822604426836505014094245058109, 14.238398922451426160824613022128, 15.08590870400923884345249006022, 15.994824512152465096197543390183, 17.893148112397461688808943307892, 18.553339257267108997700464916804, 19.5801681683445760642176141231, 21.299555848696361388853249678660, 22.66130022536784150067703846598, 23.714653280064396340068626310318, 24.50418839318763441143845825065, 25.65494043768818283766769081609, 26.42406201862116899635630125552, 27.55968075016591434278086474599, 29.047267151442792067828807261432, 30.74297617938886684124488420774, 31.286243100925370531320530639911, 31.84995154290089965468122776546

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