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Properties

Label 1-67-67.33-r0-0-0
Degree $1$
Conductor $67$
Sign $0.970 + 0.241i$
Analytic cond. $0.311146$
Root an. cond. $0.311146$
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  + (−0.995 + 0.0950i)2-s + (0.841 − 0.540i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−0.786 + 0.618i)6-s + (0.580 + 0.814i)7-s + (−0.959 + 0.281i)8-s + (0.415 − 0.909i)9-s + (0.0475 − 0.998i)10-s + (−0.786 − 0.618i)11-s + (0.723 − 0.690i)12-s + (0.235 + 0.971i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (0.928 − 0.371i)16-s + (0.981 + 0.189i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)2-s + (0.841 − 0.540i)3-s + (0.981 − 0.189i)4-s + (−0.142 + 0.989i)5-s + (−0.786 + 0.618i)6-s + (0.580 + 0.814i)7-s + (−0.959 + 0.281i)8-s + (0.415 − 0.909i)9-s + (0.0475 − 0.998i)10-s + (−0.786 − 0.618i)11-s + (0.723 − 0.690i)12-s + (0.235 + 0.971i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (0.928 − 0.371i)16-s + (0.981 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.970 + 0.241i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.970 + 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7956583073 + 0.09758800494i\)
\(L(\frac12)\) \(\approx\) \(0.7956583073 + 0.09758800494i\)
\(L(1)\) \(\approx\) \(0.8785456798 + 0.05532958117i\)
\(L(1)\) \(\approx\) \(0.8785456798 + 0.05532958117i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.995 + 0.0950i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (0.580 + 0.814i)T \)
11 \( 1 + (-0.786 - 0.618i)T \)
13 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (0.981 + 0.189i)T \)
19 \( 1 + (0.580 - 0.814i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.235 - 0.971i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.327 - 0.945i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (0.0475 + 0.998i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (0.981 - 0.189i)T \)
73 \( 1 + (-0.786 + 0.618i)T \)
79 \( 1 + (0.723 - 0.690i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
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

−32.06679588518937904287141132762, −30.79727261874905039009756283919, −29.75384790663342352682066309479, −28.279387191169992490892688409538, −27.5341437750129923817717222927, −26.65734762362093198521188283492, −25.501741963517867924165414696124, −24.6948420032269993919849294584, −23.37877600557157107514109201893, −21.25741776496837774046941932295, −20.4351798197228920619088541627, −20.04818333877437006386044525110, −18.49421529302696260611477610122, −17.18931856173526047547205583088, −16.14713277308852203127031021334, −15.18496923713957757796227273750, −13.6048806560416233572865294512, −12.13881403260718866049428599941, −10.44053052391080622439274702376, −9.70023653231159514607168483192, −8.09485955200318559897614493648, −7.77035158578022494896892067122, −5.17744442764163922219091536959, −3.48557128001119582670762552136, −1.57231307169983707910662156661, 1.97356849405276657427457640698, 3.12417026166472698082765967443, 6.02590156402367937895401964128, 7.36544403311016138469687912911, 8.29133417186128776317406875813, 9.48181550244603111256884632928, 10.974756413278443197987141744977, 12.09110094150652404830193646682, 14.01560318131744664776010514737, 14.962967925723451115892545027687, 16.04583545752882940809719088755, 17.91830571187625760223210857840, 18.613681077565109034265870155281, 19.23945907571926964710018271917, 20.7057797630188296278069070040, 21.67403433306451347102817417592, 23.75656277699205308198211539390, 24.48471624847862239394358877968, 25.92989598012372272678609588245, 26.21642551517614421915108341058, 27.48727906417679031834421960821, 28.76797443891157756181925793386, 29.93026826626426342589858897729, 30.751541866815005850046656687229, 31.83162820678277496728856065103

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