Properties

Label 2-24e2-576.517-c1-0-86
Degree $2$
Conductor $576$
Sign $0.916 + 0.400i$
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  + (1.09 + 0.890i)2-s + (0.224 − 1.71i)3-s + (0.412 + 1.95i)4-s + (2.72 − 3.10i)5-s + (1.77 − 1.68i)6-s + (2.35 − 0.310i)7-s + (−1.29 + 2.51i)8-s + (−2.89 − 0.771i)9-s + (5.75 − 0.983i)10-s + (0.107 − 0.217i)11-s + (3.45 − 0.268i)12-s + (−0.686 + 2.02i)13-s + (2.86 + 1.76i)14-s + (−4.71 − 5.37i)15-s + (−3.65 + 1.61i)16-s + (−4.79 + 4.79i)17-s + ⋯
L(s)  = 1  + (0.776 + 0.629i)2-s + (0.129 − 0.991i)3-s + (0.206 + 0.978i)4-s + (1.21 − 1.38i)5-s + (0.725 − 0.688i)6-s + (0.891 − 0.117i)7-s + (−0.456 + 0.889i)8-s + (−0.966 − 0.257i)9-s + (1.81 − 0.311i)10-s + (0.0322 − 0.0654i)11-s + (0.996 − 0.0775i)12-s + (−0.190 + 0.560i)13-s + (0.766 + 0.470i)14-s + (−1.21 − 1.38i)15-s + (−0.914 + 0.403i)16-s + (−1.16 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74168 - 0.572237i\)
\(L(\frac12)\) \(\approx\) \(2.74168 - 0.572237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 - 0.890i)T \)
3 \( 1 + (-0.224 + 1.71i)T \)
good5 \( 1 + (-2.72 + 3.10i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (-2.35 + 0.310i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (-0.107 + 0.217i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (0.686 - 2.02i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (4.79 - 4.79i)T - 17iT^{2} \)
19 \( 1 + (1.45 + 7.33i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (0.332 - 2.52i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (-4.29 - 0.281i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (-7.75 - 4.48i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.502 - 2.52i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.940 - 7.14i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (-6.11 - 3.01i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (9.30 + 2.49i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.08 - 1.63i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-1.34 + 1.53i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (0.125 - 1.92i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (-5.67 + 2.79i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (-3.51 + 8.48i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.86 - 6.91i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.38 + 12.6i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.404 + 0.355i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (8.74 + 3.62i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (5.66 - 3.27i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.00477557496813615435961631535, −9.375379753610475581320461297796, −8.536902503796069400746698448295, −8.171435163298455690771321538086, −6.72139449382849482572295512823, −6.23636085901595513854940429050, −5.02495376807834379028524048799, −4.54637957294888368123548036968, −2.51247876169906961948165277439, −1.49674133412435411153613080783, 2.16268837078490598444938077076, 2.81953375781047998078740929495, 4.08778332659391821424892063759, 5.14774920631383991073382838771, 5.91823071772998213608223256386, 6.81570471566623643192379826374, 8.302060377197106121435760755253, 9.617182948711716203744620604304, 10.04310235409577080846453307144, 10.83707766971824113295369481307

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