↑

Properties

Label 2-468-117.11-c1-0-6
Degree $2$
Conductor $468$
Sign $0.831 + 0.555i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 + 1.18i)3-s + (−1.99 − 0.535i)5-s + (−0.888 − 0.238i)7-s + (0.213 − 2.99i)9-s + (0.0757 + 0.0757i)11-s + (3.56 − 0.509i)13-s + (3.16 − 1.68i)15-s + (2.19 − 3.80i)17-s + (5.47 − 1.46i)19-s + (1.40 − 0.747i)21-s + (−1.12 + 1.95i)23-s + (−0.622 − 0.359i)25-s + (3.26 + 4.04i)27-s − 4.12i·29-s + (0.267 − 0.998i)31-s + ⋯
L(s)  = 1  + (−0.731 + 0.681i)3-s + (−0.893 − 0.239i)5-s + (−0.335 − 0.0900i)7-s + (0.0712 − 0.997i)9-s + (0.0228 + 0.0228i)11-s + (0.989 − 0.141i)13-s + (0.817 − 0.433i)15-s + (0.532 − 0.921i)17-s + (1.25 − 0.336i)19-s + (0.307 − 0.163i)21-s + (−0.235 + 0.408i)23-s + (−0.124 − 0.0718i)25-s + (0.627 + 0.778i)27-s − 0.765i·29-s + (0.0480 − 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.831 + 0.555i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.831 + 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809099 - 0.245463i\)
\(L(\frac12)\) \(\approx\) \(0.809099 - 0.245463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.26 - 1.18i)T \)
13 \( 1 + (-3.56 + 0.509i)T \)
good5 \( 1 + (1.99 + 0.535i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.888 + 0.238i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.0757 - 0.0757i)T + 11iT^{2} \)
17 \( 1 + (-2.19 + 3.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.47 + 1.46i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.12 - 1.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.12iT - 29T^{2} \)
31 \( 1 + (-0.267 + 0.998i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-8.13 - 2.17i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.749 + 2.79i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.43 + 4.87i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.24 - 0.868i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 + (6.00 + 6.00i)T + 59iT^{2} \)
61 \( 1 + (1.81 + 3.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.60 + 0.429i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.10 - 11.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.23 - 3.23i)T - 73iT^{2} \)
79 \( 1 + (0.616 - 1.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.55 - 5.80i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-4.05 + 15.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.17 + 4.38i)T + (-84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.26637540501577024219790233995, −9.966531875709289910308033699200, −9.412137756735201397792558240217, −8.219645671647268964544237499286, −7.27722242390297195618892374084, −6.13185149223079087230384548797, −5.19294718951308366805156713996, −4.10732768822964814148338541041, −3.25629687380241178807745753594, −0.68373357277270202317266558799, 1.26853876411092517733404746407, 3.14806666533872956480265624083, 4.31240179657219787782682484217, 5.71505031017910017873020068155, 6.39279647130295068666495189990, 7.55945165536927358890104357810, 8.044059768491117288289452627656, 9.322066513492769601546460734076, 10.55676400085066593675135336362, 11.17402866896440259735902906729

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