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Properties

Label 1-67-67.58-r1-0-0
Degree $1$
Conductor $67$
Sign $0.611 - 0.791i$
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  + (−0.415 − 0.909i)2-s + (−0.841 − 0.540i)3-s + (−0.654 + 0.755i)4-s + (0.142 + 0.989i)5-s + (−0.142 + 0.989i)6-s + (−0.415 − 0.909i)7-s + (0.959 + 0.281i)8-s + (0.415 + 0.909i)9-s + (0.841 − 0.540i)10-s + (0.142 + 0.989i)11-s + (0.959 − 0.281i)12-s + (0.959 − 0.281i)13-s + (−0.654 + 0.755i)14-s + (0.415 − 0.909i)15-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (−0.841 − 0.540i)3-s + (−0.654 + 0.755i)4-s + (0.142 + 0.989i)5-s + (−0.142 + 0.989i)6-s + (−0.415 − 0.909i)7-s + (0.959 + 0.281i)8-s + (0.415 + 0.909i)9-s + (0.841 − 0.540i)10-s + (0.142 + 0.989i)11-s + (0.959 − 0.281i)12-s + (0.959 − 0.281i)13-s + (−0.654 + 0.755i)14-s + (0.415 − 0.909i)15-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.611 - 0.791i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ 0.611 - 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8116531719 - 0.3987034370i\)
\(L(\frac12)\) \(\approx\) \(0.8116531719 - 0.3987034370i\)
\(L(1)\) \(\approx\) \(0.6572560285 - 0.2728139545i\)
\(L(1)\) \(\approx\) \(0.6572560285 - 0.2728139545i\)

Euler product

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

−32.40175924599560664561268468714, −31.27818735316315867352850659827, −28.9798826890412348441210562655, −28.54322543367339695371086253711, −27.554106721410904530253531466893, −26.52897632737566445349925825246, −25.16372840662106425925318692467, −24.26925701133763287875303525405, −23.235577867630861562697473078693, −22.051082489828959669311140299565, −20.972298199913033136695148691532, −19.20040334446565784444926953383, −18.0910251742620347086190020505, −16.85590071526772690580217720374, −16.16411740812216672996840084661, −15.307660529267382847648551351368, −13.5651034211689512681095056738, −12.20721726003103513663913944886, −10.6663502527538072276007592324, −9.22137294126389853142127987457, −8.4986831516969910905864156956, −6.26266012501577742139049237914, −5.64869568862312627907248781825, −4.205496132528574809974378771732, −0.89387759782224770075762029247, 0.96986378232948254441805470593, 2.77913276182525733877962302614, 4.521637013776537375455120645831, 6.63046748934134493391855797033, 7.558738651970168166263226805874, 9.64597465150373120816160606893, 10.745731703396220484131842802690, 11.4915260528231276529844982491, 12.99283871101191636647507998859, 13.81186326979104223938697532849, 15.91129148849787355940612476921, 17.48915089173315130065818512886, 17.92811992338104209192334819604, 19.1507940080459268388326788652, 20.16972878173570529543340706435, 21.66563162388794442272793741287, 22.89121532897966774566252194353, 23.12060781140854339647929272744, 25.23596558462286722889549745578, 26.32491179549182153724108969248, 27.376521137303181479975606207191, 28.57679735542988896433855206878, 29.38000213596219113568092225711, 30.368318628639455109088907723193, 30.838377722334014048467473476420

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