↑

Properties

Label 2-300-60.47-c2-0-38
Degree $2$
Conductor $300$
Sign $0.921 - 0.388i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 + 0.141i)2-s + (2.17 + 2.06i)3-s + (3.95 − 0.565i)4-s + (−4.63 − 3.81i)6-s + (5.18 − 5.18i)7-s + (−7.81 + 1.68i)8-s + (0.459 + 8.98i)9-s + 7.14·11-s + (9.78 + 6.95i)12-s + (7.93 − 7.93i)13-s + (−9.61 + 11.0i)14-s + (15.3 − 4.47i)16-s + (16.5 − 16.5i)17-s + (−2.19 − 17.8i)18-s − 12.1·19-s + ⋯
L(s)  = 1  + (−0.997 + 0.0708i)2-s + (0.724 + 0.688i)3-s + (0.989 − 0.141i)4-s + (−0.771 − 0.635i)6-s + (0.741 − 0.741i)7-s + (−0.977 + 0.211i)8-s + (0.0510 + 0.998i)9-s + 0.649·11-s + (0.815 + 0.579i)12-s + (0.610 − 0.610i)13-s + (−0.686 + 0.791i)14-s + (0.960 − 0.279i)16-s + (0.975 − 0.975i)17-s + (−0.121 − 0.992i)18-s − 0.639·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.921 - 0.388i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.921 - 0.388i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.51997 + 0.307501i\)
\(L(\frac12)\) \(\approx\) \(1.51997 + 0.307501i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.141i)T \)
3 \( 1 + (-2.17 - 2.06i)T \)
5 \( 1 \)
good7 \( 1 + (-5.18 + 5.18i)T - 49iT^{2} \)
11 \( 1 - 7.14T + 121T^{2} \)
13 \( 1 + (-7.93 + 7.93i)T - 169iT^{2} \)
17 \( 1 + (-16.5 + 16.5i)T - 289iT^{2} \)
19 \( 1 + 12.1T + 361T^{2} \)
23 \( 1 + (11.0 - 11.0i)T - 529iT^{2} \)
29 \( 1 - 26.1T + 841T^{2} \)
31 \( 1 - 8.74iT - 961T^{2} \)
37 \( 1 + (-26.7 - 26.7i)T + 1.36e3iT^{2} \)
41 \( 1 + 35.4iT - 1.68e3T^{2} \)
43 \( 1 + (24.6 + 24.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-58.6 - 58.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-20.4 - 20.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 59.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.42T + 3.72e3T^{2} \)
67 \( 1 + (-35.8 + 35.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 46.2T + 5.04e3T^{2} \)
73 \( 1 + (10.6 - 10.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 68.3T + 6.24e3T^{2} \)
83 \( 1 + (76.6 - 76.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 41.0T + 7.92e3T^{2} \)
97 \( 1 + (81.7 + 81.7i)T + 9.40e3iT^{2} \)
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

−11.18183249472054853677528763096, −10.45228566101222154662344515962, −9.706655626910110916824094978229, −8.708809099497180075304944943685, −7.971395896007856589614004378335, −7.14102424058429352346828836809, −5.64788936352329885714370040894, −4.20560998972270697318331050135, −2.91233285404451694683399148869, −1.26034448821246743534957463051, 1.32298452558132509654488655496, 2.34462280560573400377488194159, 3.84983771938791636077044155005, 5.94836843279686742476026149860, 6.74363193118396015281895963605, 8.030000245209640768770914015959, 8.475038053227313475035734932282, 9.295853638609723695193302930568, 10.38285593289097235373282351339, 11.60248811684082268842932426742

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