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Properties

Label 1-67-67.61-r1-0-0
Degree $1$
Conductor $67$
Sign $-0.788 + 0.614i$
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  + (0.786 + 0.618i)2-s + (0.654 + 0.755i)3-s + (0.235 + 0.971i)4-s + (−0.841 − 0.540i)5-s + (0.0475 + 0.998i)6-s + (−0.928 + 0.371i)7-s + (−0.415 + 0.909i)8-s + (−0.142 + 0.989i)9-s + (−0.327 − 0.945i)10-s + (−0.0475 + 0.998i)11-s + (−0.580 + 0.814i)12-s + (0.995 + 0.0950i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (−0.888 + 0.458i)16-s + (0.235 − 0.971i)17-s + ⋯
L(s)  = 1  + (0.786 + 0.618i)2-s + (0.654 + 0.755i)3-s + (0.235 + 0.971i)4-s + (−0.841 − 0.540i)5-s + (0.0475 + 0.998i)6-s + (−0.928 + 0.371i)7-s + (−0.415 + 0.909i)8-s + (−0.142 + 0.989i)9-s + (−0.327 − 0.945i)10-s + (−0.0475 + 0.998i)11-s + (−0.580 + 0.814i)12-s + (0.995 + 0.0950i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (−0.888 + 0.458i)16-s + (0.235 − 0.971i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.788 + 0.614i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ -0.788 + 0.614i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7473735628 + 2.174334476i\)
\(L(\frac12)\) \(\approx\) \(0.7473735628 + 2.174334476i\)
\(L(1)\) \(\approx\) \(1.180732710 + 1.116733277i\)
\(L(1)\) \(\approx\) \(1.180732710 + 1.116733277i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.786 + 0.618i)T \)
3 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (-0.928 + 0.371i)T \)
11 \( 1 + (-0.0475 + 0.998i)T \)
13 \( 1 + (0.995 + 0.0950i)T \)
17 \( 1 + (0.235 - 0.971i)T \)
19 \( 1 + (0.928 + 0.371i)T \)
23 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.995 - 0.0950i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.723 - 0.690i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (-0.327 + 0.945i)T \)
53 \( 1 + (0.959 + 0.281i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (0.235 + 0.971i)T \)
73 \( 1 + (0.0475 + 0.998i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (-0.888 + 0.458i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−31.17619451244693687373161446964, −30.31318807396850315822521336627, −29.54011090613473256595885659465, −28.40319737787407767540459185539, −26.81767964469241511136683436420, −25.7838093259705816781492612932, −24.34440835480608232361196543697, −23.44461619224977206206695416397, −22.63827731796494312282011370982, −21.18265710003194951432255128694, −19.8647078028959228770994517730, −19.260232225857309558365882127916, −18.396831414274326032459325089589, −16.107665752025082067793706860550, −14.99900305160895400958694906869, −13.76831364039308147331635442342, −12.97861072668227526441972323479, −11.69769821961591157061698161095, −10.53314100633785269947505681811, −8.80824521316975995866584596102, −7.16473332194451333063444726044, −6.05991538642510426018102549839, −3.66171917893408629227390398546, −3.07588557012837310608457020799, −0.91226624847186112823975507152, 2.97368497572551911186904767689, 4.08063067165637858568165521005, 5.29576070725595261294346250026, 7.14080585870521968550792098412, 8.40999824647845172528153920991, 9.57719283493888036796098246403, 11.54949521385430765652691807914, 12.76962307165165622069368773649, 13.908254223760184743994005011320, 15.44404133200868936207909828083, 15.7514862902113201301745437719, 16.85844229413294627424017969410, 18.86954279229468054321728252205, 20.35462785540722751539352567112, 20.87920071469035509674658583718, 22.549542802361487488894948198439, 22.996432421734948533278288851109, 24.63974546409513272757725531724, 25.48013309241850056452413831911, 26.429327989262434378834479757437, 27.60688738661150718783141418783, 28.81728690468238977065908634969, 30.70407738165373243452956516711, 31.259328093677800669911250667184, 32.2057139360872545917211106129

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