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Properties

Label 2-24e2-16.13-c1-0-2
Degree $2$
Conductor $576$
Sign $0.684 - 0.728i$
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  + (0.595 − 0.595i)5-s + 1.64i·7-s + (−3.36 + 3.36i)11-s + (2.64 + 2.64i)13-s + 5.53·17-s + (3.64 + 3.64i)19-s − 4.33i·23-s + 4.29i·25-s + (−6.12 − 6.12i)29-s + 5.64·31-s + (0.979 + 0.979i)35-s + (−0.645 + 0.645i)37-s + 7.91i·41-s + (0.354 − 0.354i)43-s + 9.10·47-s + ⋯
L(s)  = 1  + (0.266 − 0.266i)5-s + 0.622i·7-s + (−1.01 + 1.01i)11-s + (0.733 + 0.733i)13-s + 1.34·17-s + (0.836 + 0.836i)19-s − 0.904i·23-s + 0.858i·25-s + (−1.13 − 1.13i)29-s + 1.01·31-s + (0.165 + 0.165i)35-s + (−0.106 + 0.106i)37-s + 1.23i·41-s + (0.0540 − 0.0540i)43-s + 1.32·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.684 - 0.728i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.684 - 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36396 + 0.590172i\)
\(L(\frac12)\) \(\approx\) \(1.36396 + 0.590172i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.595 + 0.595i)T - 5iT^{2} \)
7 \( 1 - 1.64iT - 7T^{2} \)
11 \( 1 + (3.36 - 3.36i)T - 11iT^{2} \)
13 \( 1 + (-2.64 - 2.64i)T + 13iT^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 + (-3.64 - 3.64i)T + 19iT^{2} \)
23 \( 1 + 4.33iT - 23T^{2} \)
29 \( 1 + (6.12 + 6.12i)T + 29iT^{2} \)
31 \( 1 - 5.64T + 31T^{2} \)
37 \( 1 + (0.645 - 0.645i)T - 37iT^{2} \)
41 \( 1 - 7.91iT - 41T^{2} \)
43 \( 1 + (-0.354 + 0.354i)T - 43iT^{2} \)
47 \( 1 - 9.10T + 47T^{2} \)
53 \( 1 + (4.93 - 4.93i)T - 53iT^{2} \)
59 \( 1 + (4.33 - 4.33i)T - 59iT^{2} \)
61 \( 1 + (0.645 + 0.645i)T + 61iT^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + 3.29iT - 73T^{2} \)
79 \( 1 + 9.64T + 79T^{2} \)
83 \( 1 + (-3.36 - 3.36i)T + 83iT^{2} \)
89 \( 1 - 2.38iT - 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−10.74466517807524206421980206089, −9.848244060508158143828586783540, −9.250101767710808744574761788350, −8.097126629773634728186020725067, −7.44352982599689782675739496722, −6.09357830835754477503914431216, −5.39711445168551616678147569633, −4.29673911000476204701951185578, −2.90641042049311425972421644249, −1.60308240850544858424221430719, 0.935659747943139937629538796492, 2.86063787852258692871168486415, 3.67861277629623822153412791179, 5.29216931512825888082184788698, 5.83192575133362212216921080024, 7.17232825230803435887514159919, 7.87081903886004447535372326468, 8.798113313497740532864301947677, 9.941077032162618182987286606621, 10.60349095085273766123646306039

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