↑

Properties

Label 2-24e2-144.11-c1-0-10
Degree $2$
Conductor $576$
Sign $0.988 - 0.152i$
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  + (1.36 − 1.06i)3-s + (2.31 + 0.619i)5-s + (−2.51 + 4.35i)7-s + (0.733 − 2.90i)9-s + (1.03 − 0.276i)11-s + (4.00 + 1.07i)13-s + (3.81 − 1.61i)15-s + 2.22i·17-s + (0.697 + 0.697i)19-s + (1.19 + 8.62i)21-s + (2.20 − 1.27i)23-s + (0.624 + 0.360i)25-s + (−2.09 − 4.75i)27-s + (0.589 − 0.157i)29-s + (−0.190 + 0.109i)31-s + ⋯
L(s)  = 1  + (0.788 − 0.614i)3-s + (1.03 + 0.276i)5-s + (−0.949 + 1.64i)7-s + (0.244 − 0.969i)9-s + (0.310 − 0.0833i)11-s + (1.11 + 0.297i)13-s + (0.985 − 0.416i)15-s + 0.539i·17-s + (0.159 + 0.159i)19-s + (0.261 + 1.88i)21-s + (0.460 − 0.265i)23-s + (0.124 + 0.0721i)25-s + (−0.402 − 0.915i)27-s + (0.109 − 0.0293i)29-s + (−0.0342 + 0.0197i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.988 - 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12604 + 0.162763i\)
\(L(\frac12)\) \(\approx\) \(2.12604 + 0.162763i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.36 + 1.06i)T \)
good5 \( 1 + (-2.31 - 0.619i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.51 - 4.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.03 + 0.276i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-4.00 - 1.07i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 2.22iT - 17T^{2} \)
19 \( 1 + (-0.697 - 0.697i)T + 19iT^{2} \)
23 \( 1 + (-2.20 + 1.27i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.589 + 0.157i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.190 - 0.109i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.16 + 5.16i)T + 37iT^{2} \)
41 \( 1 + (-0.828 - 1.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.33 + 4.98i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.76 + 9.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.80 - 7.80i)T - 53iT^{2} \)
59 \( 1 + (1.36 - 5.09i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.73 - 6.48i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.20 + 8.22i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + (7.74 + 4.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.48 - 5.55i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 6.56T + 89T^{2} \)
97 \( 1 + (1.51 - 2.62i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.57779149013119162092042674624, −9.562816708322149897669745690809, −8.974501666449773495851486463688, −8.432159145616890517893414603480, −6.94748147643944638136706640573, −6.17296435994536708295096808382, −5.66445259359862261262785776355, −3.68415323194656433366233705650, −2.68001886727851553427406497159, −1.76174119577383532404174915540, 1.33623189626552077272387783277, 3.05291288936508082288403003598, 3.86823251393149497582285923986, 4.95841777155012389451246021568, 6.22493298205767821663857231405, 7.11813847160662692775377301435, 8.144674190716345907215007769124, 9.289995156286569025866722227273, 9.699136934436481353975839985463, 10.47938360628837269592439495782

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