↑

Properties

Label 2-30e2-45.4-c1-0-0
Degree $2$
Conductor $900$
Sign $-0.798 - 0.601i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (−1.07 − 1.36i)3-s + (−0.0748 + 0.0432i)7-s + (−0.704 + 2.91i)9-s + (0.456 + 0.791i)11-s + (−2.27 − 1.31i)13-s − 2.08i·17-s − 4.93·19-s + (0.139 + 0.0555i)21-s + (−7.34 − 4.23i)23-s + (4.72 − 2.16i)27-s + (1.19 + 2.07i)29-s + (−1.81 + 3.13i)31-s + (0.587 − 1.46i)33-s + 5.85i·37-s + (0.649 + 4.49i)39-s + ⋯
L(s)  = 1  + (−0.618 − 0.785i)3-s + (−0.0283 + 0.0163i)7-s + (−0.234 + 0.972i)9-s + (0.137 + 0.238i)11-s + (−0.630 − 0.364i)13-s − 0.506i·17-s − 1.13·19-s + (0.0303 + 0.0121i)21-s + (−1.53 − 0.883i)23-s + (0.908 − 0.416i)27-s + (0.222 + 0.385i)29-s + (−0.325 + 0.563i)31-s + (0.102 − 0.255i)33-s + 0.962i·37-s + (0.103 + 0.720i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.798 - 0.601i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.798 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00151326 + 0.00452131i\)
\(L(\frac12)\) \(\approx\) \(0.00151326 + 0.00452131i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 + 1.36i)T \)
5 \( 1 \)
good7 \( 1 + (0.0748 - 0.0432i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.456 - 0.791i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.27 + 1.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.08iT - 17T^{2} \)
19 \( 1 + 4.93T + 19T^{2} \)
23 \( 1 + (7.34 + 4.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.19 - 2.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.81 - 3.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.85iT - 37T^{2} \)
41 \( 1 + (3.32 - 5.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.14 - 4.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.32 - 1.34i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.73iT - 53T^{2} \)
59 \( 1 + (-6.16 + 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.16 - 5.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.33 - 3.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 5.31iT - 73T^{2} \)
79 \( 1 + (6.72 + 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.26 - 3.03i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.13T + 89T^{2} \)
97 \( 1 + (9.61 - 5.55i)T + (48.5 - 84.0i)T^{2} \)
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

−10.41260770161868695242197863052, −9.837324606680479628976039210077, −8.489016247252150150525616008452, −7.940702376294366795516388219747, −6.84535260574970133873010594369, −6.34290264287019132657539011178, −5.22763142487902970389269354523, −4.41626803419020543082696978689, −2.82822841264503948586131986645, −1.69610659084400319457268630598, 0.00232269997229685258244088152, 2.05204218555748909825641580736, 3.65123572169189164363314709844, 4.30542704957462967126390083043, 5.43225100126888223815728385841, 6.14642875677412872963916782048, 7.07803438870723892195930393893, 8.243732910783499275136665524093, 9.063214793950180529483576266965, 9.968603412511958114262917936391

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