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Properties

Label 2-24e2-48.35-c1-0-3
Degree $2$
Conductor $576$
Sign $0.805 - 0.592i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.63 + 2.63i)5-s + 0.207·7-s + (3.66 − 3.66i)11-s + (0.255 + 0.255i)13-s − 0.654i·17-s + (−4.46 + 4.46i)19-s + 3.48i·23-s + 8.86i·25-s + (4.33 − 4.33i)29-s + 6.16i·31-s + (0.545 + 0.545i)35-s + (4.39 − 4.39i)37-s + 0.0684·41-s + (−5.65 − 5.65i)43-s + 9.14·47-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)5-s + 0.0783·7-s + (1.10 − 1.10i)11-s + (0.0708 + 0.0708i)13-s − 0.158i·17-s + (−1.02 + 1.02i)19-s + 0.727i·23-s + 1.77i·25-s + (0.805 − 0.805i)29-s + 1.10i·31-s + (0.0921 + 0.0921i)35-s + (0.722 − 0.722i)37-s + 0.0106·41-s + (−0.862 − 0.862i)43-s + 1.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76210 + 0.577883i\)
\(L(\frac12)\) \(\approx\) \(1.76210 + 0.577883i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 2.63i)T + 5iT^{2} \)
7 \( 1 - 0.207T + 7T^{2} \)
11 \( 1 + (-3.66 + 3.66i)T - 11iT^{2} \)
13 \( 1 + (-0.255 - 0.255i)T + 13iT^{2} \)
17 \( 1 + 0.654iT - 17T^{2} \)
19 \( 1 + (4.46 - 4.46i)T - 19iT^{2} \)
23 \( 1 - 3.48iT - 23T^{2} \)
29 \( 1 + (-4.33 + 4.33i)T - 29iT^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 + (-4.39 + 4.39i)T - 37iT^{2} \)
41 \( 1 - 0.0684T + 41T^{2} \)
43 \( 1 + (5.65 + 5.65i)T + 43iT^{2} \)
47 \( 1 - 9.14T + 47T^{2} \)
53 \( 1 + (-1.51 - 1.51i)T + 53iT^{2} \)
59 \( 1 + (2.53 - 2.53i)T - 59iT^{2} \)
61 \( 1 + (5.46 + 5.46i)T + 61iT^{2} \)
67 \( 1 + (-4.77 + 4.77i)T - 67iT^{2} \)
71 \( 1 - 5.94iT - 71T^{2} \)
73 \( 1 - 6.93iT - 73T^{2} \)
79 \( 1 + 4.72iT - 79T^{2} \)
83 \( 1 + (4.32 + 4.32i)T + 83iT^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 0.925T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.74513428097726539108075795099, −10.02556356532573883947203153233, −9.176545296945595402260145125602, −8.251147789770902173128348053137, −6.95227373781996735880777347494, −6.25537762219424810226789785710, −5.62764363167535496374567842243, −3.97449486394451920625575696711, −2.92310515381225800668898613428, −1.65262804271501284425526981895, 1.27036760860143289769744904365, 2.38473701588702443560148285431, 4.32221975917231269663901501747, 4.89169046083982666715769354189, 6.13019483817472406273667326739, 6.80517013041554834130752135119, 8.229349989192815072285327004445, 9.042937790572414018702486965244, 9.593094844734981926569497791741, 10.44102390124494136892822238402

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