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Properties

Label 1-3381-3381.641-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.770 + 0.637i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.612 + 0.790i)2-s + (−0.248 + 0.968i)4-s + (−0.675 − 0.737i)5-s + (−0.917 + 0.396i)8-s + (0.169 − 0.985i)10-s + (0.906 + 0.421i)11-s + (0.970 + 0.242i)13-s + (−0.876 − 0.482i)16-s + (0.963 + 0.268i)17-s + (−0.0475 − 0.998i)19-s + (0.882 − 0.470i)20-s + (0.222 + 0.974i)22-s + (−0.0882 + 0.996i)25-s + (0.403 + 0.915i)26-s + (0.714 − 0.699i)29-s + ⋯
L(s)  = 1  + (0.612 + 0.790i)2-s + (−0.248 + 0.968i)4-s + (−0.675 − 0.737i)5-s + (−0.917 + 0.396i)8-s + (0.169 − 0.985i)10-s + (0.906 + 0.421i)11-s + (0.970 + 0.242i)13-s + (−0.876 − 0.482i)16-s + (0.963 + 0.268i)17-s + (−0.0475 − 0.998i)19-s + (0.882 − 0.470i)20-s + (0.222 + 0.974i)22-s + (−0.0882 + 0.996i)25-s + (0.403 + 0.915i)26-s + (0.714 − 0.699i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.118993152 + 0.7624528812i\)
\(L(\frac12)\) \(\approx\) \(2.118993152 + 0.7624528812i\)
\(L(1)\) \(\approx\) \(1.314428374 + 0.4827976323i\)
\(L(1)\) \(\approx\) \(1.314428374 + 0.4827976323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.612 + 0.790i)T \)
5 \( 1 + (-0.675 - 0.737i)T \)
11 \( 1 + (0.906 + 0.421i)T \)
13 \( 1 + (0.970 + 0.242i)T \)
17 \( 1 + (0.963 + 0.268i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (0.714 - 0.699i)T \)
31 \( 1 + (0.981 + 0.189i)T \)
37 \( 1 + (-0.777 + 0.628i)T \)
41 \( 1 + (0.301 - 0.953i)T \)
43 \( 1 + (-0.917 - 0.396i)T \)
47 \( 1 + (-0.365 - 0.930i)T \)
53 \( 1 + (0.314 + 0.949i)T \)
59 \( 1 + (0.169 - 0.985i)T \)
61 \( 1 + (-0.994 + 0.108i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (-0.488 - 0.872i)T \)
73 \( 1 + (-0.195 - 0.980i)T \)
79 \( 1 + (0.995 + 0.0950i)T \)
83 \( 1 + (-0.862 + 0.505i)T \)
89 \( 1 + (0.760 + 0.649i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−18.81442362941307346793798022895, −18.39704309604682272836979713308, −17.608114460125318925494872612708, −16.419116407641015485079260634, −15.93566394770070922047088933739, −15.05063361466462247819948990334, −14.36598843086955837038282758905, −14.07518680078252826542082801355, −13.13364886702946425308371383977, −12.306545531288862219448478742393, −11.69424083605410084975633209504, −11.26380884503337942328171870234, −10.35379818451434052710516039322, −9.97610373827003543600564650360, −8.84806566784244202062464950077, −8.22392809806282081893985208223, −7.22427236878133635385166115446, −6.27969260499818654541481158006, −5.92467406634838647999901564170, −4.789704791569657460734705321416, −3.96871900936653820615268859793, −3.35965556908767783595886759647, −2.8818605158497939296488961508, −1.589183796826740874876563947541, −0.90509189636233980908286738649, 0.71253497521407373221586293752, 1.78869020763746912610663931625, 3.157884270396281976657381259874, 3.74283911047641808194478175957, 4.51791295845612039072733798854, 5.042375839632211062339870430883, 6.05056141701036473563918467312, 6.666766602300105218807183094930, 7.44810549787789826347415254272, 8.234036041997917113102833004656, 8.77381279873195630196998382358, 9.41327778007070969579028108859, 10.53363111059221336080563869443, 11.67253943626822655462136528605, 11.956401206537984730966176783695, 12.61983789167295987036886955344, 13.60781014470065362003170834032, 13.850913698811528823120235612925, 14.99709391088200866966897742160, 15.38325612340551767213560695981, 16.05001847031801267609647375187, 16.79393065108325174311225116335, 17.20663030753016179792829652495, 17.98730047802923730749327350245, 18.93993308386135136968430951537

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