Properties

Label 2-24e2-8.5-c1-0-2
Degree $2$
Conductor $576$
Sign $0.707 - 0.707i$
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  + 6i·11-s + 6·17-s + 2i·19-s + 5·25-s + 6·41-s + 10i·43-s − 7·49-s + 6i·59-s − 14i·67-s + 2·73-s − 18i·83-s − 18·89-s + 10·97-s + 6i·107-s − 18·113-s + ⋯
L(s)  = 1  + 1.80i·11-s + 1.45·17-s + 0.458i·19-s + 25-s + 0.937·41-s + 1.52i·43-s − 49-s + 0.781i·59-s − 1.71i·67-s + 0.234·73-s − 1.97i·83-s − 1.90·89-s + 1.01·97-s + 0.580i·107-s − 1.69·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34360 + 0.556537i\)
\(L(\frac12)\) \(\approx\) \(1.34360 + 0.556537i\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 14iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 18iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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

−10.68875605215110782900760174035, −9.910766619397820435292696604536, −9.294410980318666161761720590765, −8.000257412159321582502579217237, −7.36817614489708322710826565227, −6.34988703323918439146387537483, −5.18506337235702411474690870015, −4.30249607173664221972382518148, −2.97330513125353271685642719044, −1.54436251318823123816344282010, 0.934200635766429061756925614761, 2.84090014861999626691144965850, 3.74346129075956270246346894895, 5.20468230002258523133212636366, 5.94302179554343314978777238889, 7.01262508502942731667609086082, 8.101683451899380447987433209410, 8.752249383741104104493204952603, 9.748629045392593465453810159587, 10.73399011300224094682780167024

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