↑

Properties

Label 2-30-3.2-c4-0-2
Degree $2$
Conductor $30$
Sign $-0.480 + 0.876i$
Analytic cond. $3.10109$
Root an. cond. $1.76099$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s + (4.32 − 7.89i)3-s − 8.00·4-s − 11.1i·5-s + (−22.3 − 12.2i)6-s − 8.05·7-s + 22.6i·8-s + (−43.5 − 68.2i)9-s − 31.6·10-s − 51.7i·11-s + (−34.5 + 63.1i)12-s + 269.·13-s + 22.7i·14-s + (−88.2 − 48.3i)15-s + 64.0·16-s + 439. i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.480 − 0.876i)3-s − 0.500·4-s − 0.447i·5-s + (−0.620 − 0.339i)6-s − 0.164·7-s + 0.353i·8-s + (−0.538 − 0.842i)9-s − 0.316·10-s − 0.427i·11-s + (−0.240 + 0.438i)12-s + 1.59·13-s + 0.116i·14-s + (−0.392 − 0.214i)15-s + 0.250·16-s + 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $-0.480 + 0.876i$
Analytic conductor: \(3.10109\)
Root analytic conductor: \(1.76099\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :2),\ -0.480 + 0.876i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.753243 - 1.27159i\)
\(L(\frac12)\) \(\approx\) \(0.753243 - 1.27159i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
3 \( 1 + (-4.32 + 7.89i)T \)
5 \( 1 + 11.1iT \)
good7 \( 1 + 8.05T + 2.40e3T^{2} \)
11 \( 1 + 51.7iT - 1.46e4T^{2} \)
13 \( 1 - 269.T + 2.85e4T^{2} \)
17 \( 1 - 439. iT - 8.35e4T^{2} \)
19 \( 1 - 529.T + 1.30e5T^{2} \)
23 \( 1 + 230. iT - 2.79e5T^{2} \)
29 \( 1 + 183. iT - 7.07e5T^{2} \)
31 \( 1 + 302.T + 9.23e5T^{2} \)
37 \( 1 + 2.36e3T + 1.87e6T^{2} \)
41 \( 1 - 883. iT - 2.82e6T^{2} \)
43 \( 1 - 2.35e3T + 3.41e6T^{2} \)
47 \( 1 - 2.46e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.76e3iT - 7.89e6T^{2} \)
59 \( 1 + 3.37e3iT - 1.21e7T^{2} \)
61 \( 1 + 391.T + 1.38e7T^{2} \)
67 \( 1 - 4.36e3T + 2.01e7T^{2} \)
71 \( 1 + 8.82e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.49e3T + 2.83e7T^{2} \)
79 \( 1 + 3.05e3T + 3.89e7T^{2} \)
83 \( 1 - 1.81e3iT - 4.74e7T^{2} \)
89 \( 1 - 7.34e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.65e4T + 8.85e7T^{2} \)
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

−15.83395958207169824654616899927, −14.15409574942127839460784459754, −13.24887668738733306391904656117, −12.27426172168566008792366738252, −10.92335370012458992341123201999, −9.119576674489834572618516020892, −8.078660932368271061174623061173, −6.04261581002991677433894238307, −3.50743145937061676681007237509, −1.30357289020205364890596209303, 3.49368414359070737381560866339, 5.35629274407396505153634719984, 7.24378523108436719890150726666, 8.810370344194197722599161596147, 9.950373949858506964797354813417, 11.41288922147883240906872052329, 13.53448768558424868756896318549, 14.30580386641118862905818411936, 15.74429162330877082738933594383, 16.07728376448860523642287600037

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