↑

Properties

Label 2-24e2-64.45-c1-0-31
Degree $2$
Conductor $576$
Sign $0.575 + 0.817i$
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  + (0.919 + 1.07i)2-s + (−0.309 + 1.97i)4-s + (−0.509 − 2.56i)5-s + (−1.78 − 4.31i)7-s + (−2.40 + 1.48i)8-s + (2.28 − 2.90i)10-s + (0.337 + 0.225i)11-s + (0.558 − 2.80i)13-s + (2.99 − 5.88i)14-s + (−3.80 − 1.22i)16-s + (−2.50 − 2.50i)17-s + (−2.54 − 0.506i)19-s + (5.22 − 0.213i)20-s + (0.0678 + 0.569i)22-s + (2.78 + 1.15i)23-s + ⋯
L(s)  = 1  + (0.650 + 0.759i)2-s + (−0.154 + 0.987i)4-s + (−0.228 − 1.14i)5-s + (−0.675 − 1.63i)7-s + (−0.851 + 0.524i)8-s + (0.722 − 0.918i)10-s + (0.101 + 0.0679i)11-s + (0.154 − 0.778i)13-s + (0.799 − 1.57i)14-s + (−0.951 − 0.306i)16-s + (−0.607 − 0.607i)17-s + (−0.584 − 0.116i)19-s + (1.16 − 0.0476i)20-s + (0.0144 + 0.121i)22-s + (0.580 + 0.240i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.575 + 0.817i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.575 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27251 - 0.660494i\)
\(L(\frac12)\) \(\approx\) \(1.27251 - 0.660494i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.919 - 1.07i)T \)
3 \( 1 \)
good5 \( 1 + (0.509 + 2.56i)T + (-4.61 + 1.91i)T^{2} \)
7 \( 1 + (1.78 + 4.31i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.337 - 0.225i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (-0.558 + 2.80i)T + (-12.0 - 4.97i)T^{2} \)
17 \( 1 + (2.50 + 2.50i)T + 17iT^{2} \)
19 \( 1 + (2.54 + 0.506i)T + (17.5 + 7.27i)T^{2} \)
23 \( 1 + (-2.78 - 1.15i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-4.40 + 2.94i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 - 0.289iT - 31T^{2} \)
37 \( 1 + (-1.93 + 0.384i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-5.97 - 2.47i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (3.47 - 5.19i)T + (-16.4 - 39.7i)T^{2} \)
47 \( 1 + (0.140 + 0.140i)T + 47iT^{2} \)
53 \( 1 + (0.438 + 0.292i)T + (20.2 + 48.9i)T^{2} \)
59 \( 1 + (-1.05 - 5.31i)T + (-54.5 + 22.5i)T^{2} \)
61 \( 1 + (-4.77 - 7.14i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (-2.53 - 3.79i)T + (-25.6 + 61.8i)T^{2} \)
71 \( 1 + (5.35 + 12.9i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.89 + 14.2i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.17 - 4.17i)T - 79iT^{2} \)
83 \( 1 + (7.83 + 1.55i)T + (76.6 + 31.7i)T^{2} \)
89 \( 1 + (-6.46 + 2.67i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 16.2iT - 97T^{2} \)
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

−10.61619131610242477808762443540, −9.546445028811446465312650252184, −8.629316102157809383228091805883, −7.76216266578586012600251841441, −6.98497958447426302936666904149, −6.06399758897815333319158800627, −4.75970768810336153596363197335, −4.26334178318478338193067240974, −3.10155763272937264344802504751, −0.64910947268555341434197516545, 2.15478901697543003945486408019, 2.92311106222334297438484396585, 3.96715380603846047982181459070, 5.29160083764666896213394549125, 6.36418067706672476741479288043, 6.73446938380577870109704495221, 8.554564034348864133875787690419, 9.220725280433813401036661896885, 10.19938639186472662032484009082, 11.04059784157824493342110002387

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