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Properties

Label 2-24e2-144.11-c1-0-1
Degree $2$
Conductor $576$
Sign $-0.958 + 0.285i$
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  + (−0.0841 + 1.73i)3-s + (−1.17 − 0.315i)5-s + (−1.93 + 3.35i)7-s + (−2.98 − 0.291i)9-s + (2.53 − 0.678i)11-s + (−2.21 − 0.594i)13-s + (0.645 − 2.01i)15-s − 1.65i·17-s + (−2.32 − 2.32i)19-s + (−5.64 − 3.63i)21-s + (−6.27 + 3.62i)23-s + (−3.03 − 1.75i)25-s + (0.754 − 5.14i)27-s + (5.23 − 1.40i)29-s + (−6.44 + 3.72i)31-s + ⋯
L(s)  = 1  + (−0.0485 + 0.998i)3-s + (−0.527 − 0.141i)5-s + (−0.732 + 1.26i)7-s + (−0.995 − 0.0970i)9-s + (0.763 − 0.204i)11-s + (−0.615 − 0.164i)13-s + (0.166 − 0.519i)15-s − 0.401i·17-s + (−0.532 − 0.532i)19-s + (−1.23 − 0.793i)21-s + (−1.30 + 0.755i)23-s + (−0.607 − 0.350i)25-s + (0.145 − 0.989i)27-s + (0.972 − 0.260i)29-s + (−1.15 + 0.668i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.958 + 0.285i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0624425 - 0.427722i\)
\(L(\frac12)\) \(\approx\) \(0.0624425 - 0.427722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0841 - 1.73i)T \)
good5 \( 1 + (1.17 + 0.315i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.93 - 3.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.53 + 0.678i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.21 + 0.594i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 1.65iT - 17T^{2} \)
19 \( 1 + (2.32 + 2.32i)T + 19iT^{2} \)
23 \( 1 + (6.27 - 3.62i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.23 + 1.40i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (6.44 - 3.72i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.499 - 0.499i)T + 37iT^{2} \)
41 \( 1 + (-5.22 - 9.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.07 + 7.72i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.91 - 3.31i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.69 - 4.69i)T - 53iT^{2} \)
59 \( 1 + (1.60 - 5.98i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.01 + 7.52i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.58 - 13.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + (-7.83 - 4.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.746 + 2.78i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 4.91T + 89T^{2} \)
97 \( 1 + (7.00 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.31208194656158929606144626786, −10.14820610612723392145642987235, −9.408402048176214714266878880438, −8.832907814529369688067428577078, −7.87080378107612858184365542796, −6.44407749101324640299975753776, −5.67469772871714528685901921070, −4.60789796188418819902015140226, −3.58673892133125130211903365831, −2.52782443306020879267012472381, 0.23296882032891396762458287292, 1.89601041818759170306349318602, 3.48605162205597479527556840550, 4.32829433137565149773796109747, 6.00857912828919578147169675774, 6.73210127153614009275959602608, 7.47915721166920886596866286533, 8.186802153787452395839697997531, 9.400393879575148850058788508786, 10.32633823029238291441987758753

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