↑

Properties

Label 2-1350-45.34-c1-0-11
Degree $2$
Conductor $1350$
Sign $-0.396 + 0.918i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−2.92 − 1.68i)7-s − 0.999i·8-s + (−2.18 + 3.78i)11-s + (5.84 − 3.37i)13-s + (1.68 + 2.92i)14-s + (−0.5 + 0.866i)16-s + 1.62i·17-s + 2.37·19-s + (3.78 − 2.18i)22-s + (1.18 − 0.686i)23-s − 6.74·26-s − 3.37i·28-s + (−0.686 + 1.18i)29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.10 − 0.637i)7-s − 0.353i·8-s + (−0.659 + 1.14i)11-s + (1.61 − 0.935i)13-s + (0.450 + 0.780i)14-s + (−0.125 + 0.216i)16-s + 0.394i·17-s + 0.544·19-s + (0.807 − 0.466i)22-s + (0.247 − 0.143i)23-s − 1.32·26-s − 0.637i·28-s + (−0.127 + 0.220i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.396 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7656513427\)
\(L(\frac12)\) \(\approx\) \(0.7656513427\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (2.92 + 1.68i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.84 + 3.37i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.62iT - 17T^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 + (-1.18 + 0.686i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.686 - 1.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.37 + 4.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.87 + 2.81i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.38 + 3.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + (2.18 + 3.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.05 + 7.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 - 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 3.11iT - 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.38 + 3.68i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (-7.25 - 4.18i)T + (48.5 + 84.0i)T^{2} \)
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

−9.540503458130993631069642716080, −8.581497472813414453638447199030, −7.80661283901616736906987636358, −7.01819012798363491601364843647, −6.23750684900839095335884351432, −5.17529446929119587422088652022, −3.79771100643049636755936628374, −3.24721073295966713832437228983, −1.87143756316764391814557276019, −0.42388302104703337868282555766, 1.23346795810376610003134818824, 2.81727445299716792283140930094, 3.57511138603100176293555448790, 5.06641062889107834321772930838, 6.09841692627271602605169620857, 6.37009971415179872518404538353, 7.47772024112494379442642114564, 8.478711832869416100401161132546, 8.957767533421267153576023929755, 9.640086828392089692666002263279

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