Properties

Label 2-24e2-576.517-c1-0-21
Degree $2$
Conductor $576$
Sign $-0.972 - 0.234i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  + (−0.832 + 1.14i)2-s + (−0.342 + 1.69i)3-s + (−0.613 − 1.90i)4-s + (1.06 − 1.20i)5-s + (−1.65 − 1.80i)6-s + (0.0200 − 0.00264i)7-s + (2.68 + 0.882i)8-s + (−2.76 − 1.16i)9-s + (0.499 + 2.21i)10-s + (0.486 − 0.986i)11-s + (3.44 − 0.391i)12-s + (−1.37 + 4.03i)13-s + (−0.0137 + 0.0251i)14-s + (1.68 + 2.21i)15-s + (−3.24 + 2.33i)16-s + (−2.58 + 2.58i)17-s + ⋯
L(s)  = 1  + (−0.588 + 0.808i)2-s + (−0.197 + 0.980i)3-s + (−0.306 − 0.951i)4-s + (0.474 − 0.540i)5-s + (−0.676 − 0.736i)6-s + (0.00759 − 0.000999i)7-s + (0.950 + 0.312i)8-s + (−0.921 − 0.387i)9-s + (0.157 + 0.701i)10-s + (0.146 − 0.297i)11-s + (0.993 − 0.112i)12-s + (−0.380 + 1.11i)13-s + (−0.00366 + 0.00672i)14-s + (0.436 + 0.571i)15-s + (−0.811 + 0.584i)16-s + (−0.625 + 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.972 - 0.234i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.972 - 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0898169 + 0.756653i\)
\(L(\frac12)\) \(\approx\) \(0.0898169 + 0.756653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.832 - 1.14i)T \)
3 \( 1 + (0.342 - 1.69i)T \)
good5 \( 1 + (-1.06 + 1.20i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (-0.0200 + 0.00264i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (-0.486 + 0.986i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (1.37 - 4.03i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (2.58 - 2.58i)T - 17iT^{2} \)
19 \( 1 + (-0.725 - 3.64i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (0.902 - 6.85i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (8.38 + 0.549i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (-4.82 - 2.78i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.768 - 3.86i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.345 + 2.62i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-11.4 - 5.63i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (-0.949 - 0.254i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.06 + 7.57i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-4.48 + 5.11i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (-0.656 + 10.0i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (7.84 - 3.86i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (0.699 - 1.68i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.18 - 5.27i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.72 + 6.44i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (2.44 - 2.14i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (-5.43 - 2.24i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-2.86 + 1.65i)T + (48.5 - 84.0i)T^{2} \)
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

−10.97471985894241435793268996301, −9.821498749042149067581958404665, −9.433781446739088320708661539510, −8.712417138264589116341241520043, −7.70913148832596717126435128969, −6.44930701981105227726249983665, −5.67949404383074030028645446363, −4.83805227799114453567723832838, −3.80471681545325876989453003608, −1.69288236732809748951351813029, 0.53328008861395393944509965179, 2.23874889405235183561161914218, 2.85123237335628523111952256027, 4.55425848640737441113883858441, 5.88948601810706196591746450877, 6.98626351233124933090948769793, 7.61552949618788655928891271734, 8.624561253922471509615168210729, 9.497456231177621947214904231636, 10.53345389771334107246759570804

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