↑

Properties

Label 2-1350-45.23-c1-0-6
Degree $2$
Conductor $1350$
Sign $-0.695 - 0.718i$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (4.57 + 1.22i)7-s + (0.707 − 0.707i)8-s + (−3 + 1.73i)11-s + (−4.57 + 1.22i)13-s + (−2.36 + 4.09i)14-s + (0.500 + 0.866i)16-s + 3.19i·19-s + (−0.896 − 3.34i)22-s + (−0.568 − 2.12i)23-s − 4.73i·26-s + (−3.34 − 3.34i)28-s + (5.36 + 9.29i)29-s + (−0.0980 + 0.169i)31-s + (−0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (1.72 + 0.462i)7-s + (0.249 − 0.249i)8-s + (−0.904 + 0.522i)11-s + (−1.26 + 0.339i)13-s + (−0.632 + 1.09i)14-s + (0.125 + 0.216i)16-s + 0.733i·19-s + (−0.191 − 0.713i)22-s + (−0.118 − 0.442i)23-s − 0.928i·26-s + (−0.632 − 0.632i)28-s + (0.996 + 1.72i)29-s + (−0.0176 + 0.0305i)31-s + (−0.170 + 0.0457i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.695 - 0.718i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.289887239\)
\(L(\frac12)\) \(\approx\) \(1.289887239\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-4.57 - 1.22i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.57 - 1.22i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 3.19iT - 19T^{2} \)
23 \( 1 + (0.568 + 2.12i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-5.36 - 9.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0980 - 0.169i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.79 - 5.79i)T - 37iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.120 + 0.448i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.55 + 5.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.79 + 5.79i)T - 53iT^{2} \)
59 \( 1 + (2.76 - 4.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.43 - 5.34i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (-3.67 - 3.67i)T + 73iT^{2} \)
79 \( 1 + (8.66 - 5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (16.6 + 4.45i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 8.66T + 89T^{2} \)
97 \( 1 + (-2.56 - 0.688i)T + (84.0 + 48.5i)T^{2} \)
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

−9.999684522541186262572509736882, −8.697810977162311066465778376345, −8.366783066614527857025320101457, −7.45914399466855304531253593231, −6.90997625018895852154308622802, −5.48443287342639377330246152216, −5.06069771186320113064802457701, −4.32508396274092804054706507009, −2.60893717886503312234611264071, −1.56044899418958829376693773538, 0.56791540040831492613850238858, 1.98789598813987747062807241495, 2.84460850387466764855912194164, 4.26277771869616908657097817613, 4.89271212656695036243638718297, 5.67241673905156670340983969514, 7.23574069051011321896221963541, 7.84002173695508956111665986442, 8.394732773337675121246405703656, 9.409708821251070969425076200341

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