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Properties

Label 2-24e2-144.85-c1-0-14
Degree $2$
Conductor $576$
Sign $0.919 + 0.392i$
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  + (−1.08 + 1.34i)3-s + (2.90 − 0.777i)5-s + (1.04 − 0.603i)7-s + (−0.636 − 2.93i)9-s + (1.36 − 5.09i)11-s + (−0.541 − 2.02i)13-s + (−2.10 + 4.75i)15-s − 3.20·17-s + (1.87 − 1.87i)19-s + (−0.322 + 2.06i)21-s + (−3.61 − 2.08i)23-s + (3.47 − 2.00i)25-s + (4.64 + 2.32i)27-s + (7.94 + 2.12i)29-s + (−1.39 + 2.41i)31-s + ⋯
L(s)  = 1  + (−0.627 + 0.778i)3-s + (1.29 − 0.347i)5-s + (0.395 − 0.228i)7-s + (−0.212 − 0.977i)9-s + (0.411 − 1.53i)11-s + (−0.150 − 0.560i)13-s + (−0.543 + 1.22i)15-s − 0.777·17-s + (0.430 − 0.430i)19-s + (−0.0703 + 0.450i)21-s + (−0.754 − 0.435i)23-s + (0.695 − 0.401i)25-s + (0.894 + 0.448i)27-s + (1.47 + 0.395i)29-s + (−0.249 + 0.432i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.919 + 0.392i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46054 - 0.298291i\)
\(L(\frac12)\) \(\approx\) \(1.46054 - 0.298291i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 1.34i)T \)
good5 \( 1 + (-2.90 + 0.777i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.04 + 0.603i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.36 + 5.09i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.541 + 2.02i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 + (-1.87 + 1.87i)T - 19iT^{2} \)
23 \( 1 + (3.61 + 2.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.94 - 2.12i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (1.39 - 2.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.10 + 5.10i)T + 37iT^{2} \)
41 \( 1 + (-9.93 - 5.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.293 + 1.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.84 - 3.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.613 + 0.613i)T + 53iT^{2} \)
59 \( 1 + (-11.8 + 3.16i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.40 + 1.98i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.64 - 9.86i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.8iT - 71T^{2} \)
73 \( 1 - 2.87iT - 73T^{2} \)
79 \( 1 + (-0.913 - 1.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.82 - 0.757i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.11iT - 89T^{2} \)
97 \( 1 + (3.06 + 5.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.70829651666071889833725758057, −9.847480893837164220256506256937, −9.051353460700051119205193712049, −8.340011641576624657096955199930, −6.70953657324495406519199643196, −5.89505018679012139979426124602, −5.25893180148768949350414591668, −4.18225227604962924460639312852, −2.81009293615830321279828262993, −0.990283385904993144888924796810, 1.71103661704052578172617907889, 2.30864923359962904530742904869, 4.43193865423128857027698851457, 5.39982197153096408943485901652, 6.34980266396897355565137489530, 6.94711901782343015277147064663, 7.942571515384934858948639149491, 9.205258269471113406290838563042, 9.967267032772429009404471817651, 10.71914844350670578279557620713

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