↑

Properties

Label 2-24e2-64.61-c1-0-37
Degree $2$
Conductor $576$
Sign $-0.804 - 0.594i$
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  + (0.887 − 1.10i)2-s + (−0.423 − 1.95i)4-s + (−0.649 − 0.434i)5-s + (−3.64 + 1.50i)7-s + (−2.52 − 1.26i)8-s + (−1.05 + 0.329i)10-s + (−5.80 + 1.15i)11-s + (2.03 − 1.35i)13-s + (−1.57 + 5.35i)14-s + (−3.64 + 1.65i)16-s + (−0.960 − 0.960i)17-s + (0.435 + 0.652i)19-s + (−0.573 + 1.45i)20-s + (−3.88 + 7.41i)22-s + (−0.421 + 1.01i)23-s + ⋯
L(s)  = 1  + (0.627 − 0.778i)2-s + (−0.211 − 0.977i)4-s + (−0.290 − 0.194i)5-s + (−1.37 + 0.570i)7-s + (−0.893 − 0.448i)8-s + (−0.333 + 0.104i)10-s + (−1.74 + 0.347i)11-s + (0.564 − 0.377i)13-s + (−0.420 + 1.43i)14-s + (−0.910 + 0.413i)16-s + (−0.232 − 0.232i)17-s + (0.100 + 0.149i)19-s + (−0.128 + 0.325i)20-s + (−0.827 + 1.58i)22-s + (−0.0879 + 0.212i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.804 - 0.594i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.804 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134276 + 0.407784i\)
\(L(\frac12)\) \(\approx\) \(0.134276 + 0.407784i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.887 + 1.10i)T \)
3 \( 1 \)
good5 \( 1 + (0.649 + 0.434i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (3.64 - 1.50i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (5.80 - 1.15i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-2.03 + 1.35i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (0.960 + 0.960i)T + 17iT^{2} \)
19 \( 1 + (-0.435 - 0.652i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (0.421 - 1.01i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.43 - 0.286i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + 6.88iT - 31T^{2} \)
37 \( 1 + (1.07 - 1.61i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-2.79 + 6.74i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.20 + 6.04i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (-6.30 - 6.30i)T + 47iT^{2} \)
53 \( 1 + (10.2 - 2.04i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (4.21 + 2.81i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (2.08 - 10.4i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-1.42 + 7.14i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (2.49 - 1.03i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.90 + 0.789i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (6.20 - 6.20i)T - 79iT^{2} \)
83 \( 1 + (1.13 + 1.70i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (1.15 + 2.79i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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

−10.26873638586136040061970111650, −9.617687507074590964487786450804, −8.640340381160337451437287851591, −7.48385034502134354045587821085, −6.15983773890059342460172338473, −5.54682625961631090273164390348, −4.37634076324167699036316279513, −3.18944625202923568709583215936, −2.38671860345199976218208703766, −0.18214370664240252148386430562, 2.87728784929309552587511624227, 3.59977209251308022596440015249, 4.80855247188213845135503656985, 5.90156871884652559966377651373, 6.69438003427274449004719930964, 7.52400660952769591466953930403, 8.353830017130120508038695241499, 9.397939412151481579431705231972, 10.43392414777261854425208793964, 11.23103756845913112648760746971

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