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Properties

Label 2-24e2-48.35-c1-0-2
Degree $2$
Conductor $576$
Sign $0.476 - 0.879i$
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  + (0.763 + 0.763i)5-s − 1.33·7-s + (−1.95 + 1.95i)11-s + (4.18 + 4.18i)13-s + 4.03i·17-s + (4.26 − 4.26i)19-s + 8.86i·23-s − 3.83i·25-s + (1.23 − 1.23i)29-s + 2.87i·31-s + (−1.02 − 1.02i)35-s + (0.434 − 0.434i)37-s − 7.81·41-s + (5.49 + 5.49i)43-s + 3.20·47-s + ⋯
L(s)  = 1  + (0.341 + 0.341i)5-s − 0.505·7-s + (−0.590 + 0.590i)11-s + (1.16 + 1.16i)13-s + 0.978i·17-s + (0.979 − 0.979i)19-s + 1.84i·23-s − 0.766i·25-s + (0.230 − 0.230i)29-s + 0.516i·31-s + (−0.172 − 0.172i)35-s + (0.0714 − 0.0714i)37-s − 1.21·41-s + (0.838 + 0.838i)43-s + 0.467·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.476 - 0.879i$
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: \((2,\ 576,\ (\ :1/2),\ 0.476 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21152 + 0.721742i\)
\(L(\frac12)\) \(\approx\) \(1.21152 + 0.721742i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.763 - 0.763i)T + 5iT^{2} \)
7 \( 1 + 1.33T + 7T^{2} \)
11 \( 1 + (1.95 - 1.95i)T - 11iT^{2} \)
13 \( 1 + (-4.18 - 4.18i)T + 13iT^{2} \)
17 \( 1 - 4.03iT - 17T^{2} \)
19 \( 1 + (-4.26 + 4.26i)T - 19iT^{2} \)
23 \( 1 - 8.86iT - 23T^{2} \)
29 \( 1 + (-1.23 + 1.23i)T - 29iT^{2} \)
31 \( 1 - 2.87iT - 31T^{2} \)
37 \( 1 + (-0.434 + 0.434i)T - 37iT^{2} \)
41 \( 1 + 7.81T + 41T^{2} \)
43 \( 1 + (-5.49 - 5.49i)T + 43iT^{2} \)
47 \( 1 - 3.20T + 47T^{2} \)
53 \( 1 + (-4.06 - 4.06i)T + 53iT^{2} \)
59 \( 1 + (-4.71 + 4.71i)T - 59iT^{2} \)
61 \( 1 + (-3.26 - 3.26i)T + 61iT^{2} \)
67 \( 1 + (-5.44 + 5.44i)T - 67iT^{2} \)
71 \( 1 + 3.76iT - 71T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + (9.73 + 9.73i)T + 83iT^{2} \)
89 \( 1 + 1.64T + 89T^{2} \)
97 \( 1 + 5.70T + 97T^{2} \)
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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

−10.88500499038467606028770037013, −9.910096057187561501087062896897, −9.286575312636751924817480256932, −8.258169110385956317245350189957, −7.15537549549348505375118822574, −6.40662568006897054767172923673, −5.44337495921909419547622087232, −4.17350370425324077752246405196, −3.06780412108495106542719462735, −1.66486649972287231918797677237, 0.858138086713066135595966079537, 2.73057382022197106082416399471, 3.70616600024428467634681135071, 5.24283342773419338262544806269, 5.82138537160192345350496636178, 6.93531043538909301303980190737, 8.092439993571082205100585128391, 8.721033732680628568087795601200, 9.811611019667721683351726518230, 10.47243967916718110885724705545

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