↑

Properties

Label 2-24e2-12.11-c1-0-4
Degree $2$
Conductor $576$
Sign $0.816 + 0.577i$
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  + 1.41i·5-s − 4i·7-s + 5.65·11-s − 4·13-s − 4.24i·17-s + 5.65·23-s + 2.99·25-s − 1.41i·29-s − 4i·31-s + 5.65·35-s + 6·37-s + 9.89i·41-s − 8i·43-s + 5.65·47-s − 9·49-s + ⋯
L(s)  = 1  + 0.632i·5-s − 1.51i·7-s + 1.70·11-s − 1.10·13-s − 1.02i·17-s + 1.17·23-s + 0.599·25-s − 0.262i·29-s − 0.718i·31-s + 0.956·35-s + 0.986·37-s + 1.54i·41-s − 1.21i·43-s + 0.825·47-s − 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44379 - 0.458892i\)
\(L(\frac12)\) \(\approx\) \(1.44379 - 0.458892i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 9.89iT - 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 - 4.24iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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

−10.69619063980368636171375991177, −9.733462023742142104423597225146, −9.147631667986383707719935815804, −7.64006565909740857192430892332, −7.07012598023383490879371606100, −6.41109217734208565932017327079, −4.81626122090875509237041380955, −3.98491503100421461092184165467, −2.82630491320167199391182977328, −1.00139004576048210737172999504, 1.51386925148197022456047979947, 2.87620669905022926389970862274, 4.30200732071547033856468519348, 5.27264475672384624893226264143, 6.18668713965497884219095361266, 7.15114571207234710370819670074, 8.521256754362502619770605852771, 8.988662126596628266130241397373, 9.636746367279832582119890311446, 10.92295975230878776874046135427

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