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Properties

Label 2-24e2-576.155-c1-0-19
Degree $2$
Conductor $576$
Sign $-0.996 - 0.0849i$
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  + (1.18 + 0.767i)2-s + (−1.59 + 0.685i)3-s + (0.823 + 1.82i)4-s + (0.821 + 2.41i)5-s + (−2.41 − 0.406i)6-s + (−2.87 + 2.20i)7-s + (−0.420 + 2.79i)8-s + (2.06 − 2.17i)9-s + (−0.880 + 3.50i)10-s + (0.0111 + 0.169i)11-s + (−2.55 − 2.33i)12-s + (2.52 − 2.21i)13-s + (−5.10 + 0.415i)14-s + (−2.96 − 3.28i)15-s + (−2.64 + 3.00i)16-s + (−2.69 − 2.69i)17-s + ⋯
L(s)  = 1  + (0.840 + 0.542i)2-s + (−0.918 + 0.395i)3-s + (0.411 + 0.911i)4-s + (0.367 + 1.08i)5-s + (−0.986 − 0.165i)6-s + (−1.08 + 0.832i)7-s + (−0.148 + 0.988i)8-s + (0.687 − 0.726i)9-s + (−0.278 + 1.10i)10-s + (0.00334 + 0.0511i)11-s + (−0.738 − 0.674i)12-s + (0.699 − 0.613i)13-s + (−1.36 + 0.110i)14-s + (−0.765 − 0.848i)15-s + (−0.661 + 0.750i)16-s + (−0.654 − 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.996 - 0.0849i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.996 - 0.0849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0613664 + 1.44231i\)
\(L(\frac12)\) \(\approx\) \(0.0613664 + 1.44231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 - 0.767i)T \)
3 \( 1 + (1.59 - 0.685i)T \)
good5 \( 1 + (-0.821 - 2.41i)T + (-3.96 + 3.04i)T^{2} \)
7 \( 1 + (2.87 - 2.20i)T + (1.81 - 6.76i)T^{2} \)
11 \( 1 + (-0.0111 - 0.169i)T + (-10.9 + 1.43i)T^{2} \)
13 \( 1 + (-2.52 + 2.21i)T + (1.69 - 12.8i)T^{2} \)
17 \( 1 + (2.69 + 2.69i)T + 17iT^{2} \)
19 \( 1 + (0.102 + 0.516i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (3.32 + 2.55i)T + (5.95 + 22.2i)T^{2} \)
29 \( 1 + (-2.28 - 4.64i)T + (-17.6 + 23.0i)T^{2} \)
31 \( 1 + (-4.64 - 8.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.76 + 0.549i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (1.55 - 2.02i)T + (-10.6 - 39.6i)T^{2} \)
43 \( 1 + (0.0771 + 1.17i)T + (-42.6 + 5.61i)T^{2} \)
47 \( 1 + (-2.09 - 7.83i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.57 - 6.84i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-8.38 + 2.84i)T + (46.8 - 35.9i)T^{2} \)
61 \( 1 + (5.75 + 11.6i)T + (-37.1 + 48.3i)T^{2} \)
67 \( 1 + (-0.906 + 13.8i)T + (-66.4 - 8.74i)T^{2} \)
71 \( 1 + (0.271 - 0.655i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-5.69 - 13.7i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.22 - 4.56i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (1.65 - 4.87i)T + (-65.8 - 50.5i)T^{2} \)
89 \( 1 + (6.06 - 14.6i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (0.474 + 0.273i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.10080964997650928721072722766, −10.54219919895273148393296074392, −9.520704057732668493083036630314, −8.459558991256066868965502933810, −6.91325133085388630127218507562, −6.51654733554919137718647120768, −5.81839056937741779195343899144, −4.84823381465646314133568323792, −3.48974599476016663619165586068, −2.67925971226064089508255405609, 0.68873059417771278874238371789, 1.95186209096870306392165063083, 3.84769710941058737766062638296, 4.52692327457792499644455546060, 5.76478108811762783762248885561, 6.28653010535756384752265738800, 7.20743469794875605934246459879, 8.687438999121378267703298230566, 9.888230573646964589728631303671, 10.30227553468309326264698070775

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