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Properties

Label 1-1512-1512.1355-r0-0-0
Degree $1$
Conductor $1512$
Sign $0.762 - 0.646i$
Analytic cond. $7.02169$
Root an. cond. $7.02169$
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  + (0.173 + 0.984i)5-s + (−0.173 + 0.984i)11-s + (−0.173 − 0.984i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.173 − 0.984i)29-s + (−0.173 − 0.984i)31-s − 37-s + (−0.173 − 0.984i)41-s + (0.766 + 0.642i)43-s + (0.173 − 0.984i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)5-s + (−0.173 + 0.984i)11-s + (−0.173 − 0.984i)13-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (0.173 − 0.984i)29-s + (−0.173 − 0.984i)31-s − 37-s + (−0.173 − 0.984i)41-s + (0.766 + 0.642i)43-s + (0.173 − 0.984i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(7.02169\)
Root analytic conductor: \(7.02169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1512,\ (0:\ ),\ 0.762 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.269540774 - 0.4658149224i\)
\(L(\frac12)\) \(\approx\) \(1.269540774 - 0.4658149224i\)
\(L(1)\) \(\approx\) \(1.048088638 + 0.0008906901670i\)
\(L(1)\) \(\approx\) \(1.048088638 + 0.0008906901670i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.173 + 0.984i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.173 + 0.984i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−20.84656452212509215069649264296, −19.91572777662454073985741324326, −19.13635832652069381820286764715, −18.6907106432441545429747250553, −17.42348167910725903714776712055, −16.96782061596324442908186632476, −16.2444308378700263569689194988, −15.66412946800710408193136010639, −14.40417272515778059791983110933, −14.025867455246952425539872865037, −12.936662062544194787417380936353, −12.497482850750580263737205744500, −11.59431827816837116338838317022, −10.75368677408777791904947080662, −9.90291181887435003002072472121, −8.91703635318386650055533652070, −8.537490272652216710228859073852, −7.58862879436747257106592189317, −6.51241156223185124911605696606, −5.694273062948353410106967346564, −4.97883355359778900944701294955, −4.00313202075243756583768046971, −3.191990608641773455692636267938, −1.81821572698956595753381657458, −1.14230380382901233998296842914, 0.54477924497645079523331064157, 2.1705221649082810431901378488, 2.68864063653367702541407144969, 3.68769609128993518445756245796, 4.77421201054782602800691521421, 5.555356534512517751524933905138, 6.59856125568075837919455866107, 7.243256387330561448520382533804, 7.9177365961922980958989232963, 9.07345377881327066764089719218, 9.9278141751035914503446420987, 10.47680112979453054385320525529, 11.29016967575253237451352601513, 12.141166571322913273738124935228, 13.00232299645422263833391333659, 13.70910457867470146565863643957, 14.66652736615916258148370715295, 15.1709960483455496858381773289, 15.78424187145912225201926410459, 17.003677000240644997468206728395, 17.60544861258715632575897770709, 18.22929028382506149463708762575, 19.00152395679924861542723770092, 19.689280129825318544155220578179, 20.68125758598216163135204521596

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