Properties

Label 2-24e2-576.205-c1-0-69
Degree $2$
Conductor $576$
Sign $-0.624 + 0.780i$
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  + (−1.25 − 0.650i)2-s + (1.31 − 1.12i)3-s + (1.15 + 1.63i)4-s + (−0.687 − 0.783i)5-s + (−2.38 + 0.560i)6-s + (0.664 + 0.0874i)7-s + (−0.388 − 2.80i)8-s + (0.460 − 2.96i)9-s + (0.353 + 1.43i)10-s + (−0.920 − 1.86i)11-s + (3.35 + 0.847i)12-s + (0.0795 + 0.234i)13-s + (−0.777 − 0.541i)14-s + (−1.78 − 0.256i)15-s + (−1.33 + 3.77i)16-s + (−1.10 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.888 − 0.459i)2-s + (0.759 − 0.650i)3-s + (0.577 + 0.816i)4-s + (−0.307 − 0.350i)5-s + (−0.973 + 0.228i)6-s + (0.251 + 0.0330i)7-s + (−0.137 − 0.990i)8-s + (0.153 − 0.988i)9-s + (0.111 + 0.452i)10-s + (−0.277 − 0.562i)11-s + (0.969 + 0.244i)12-s + (0.0220 + 0.0649i)13-s + (−0.207 − 0.144i)14-s + (−0.461 − 0.0661i)15-s + (−0.333 + 0.942i)16-s + (−0.268 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.624 + 0.780i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.624 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465362 - 0.968561i\)
\(L(\frac12)\) \(\approx\) \(0.465362 - 0.968561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 + 0.650i)T \)
3 \( 1 + (-1.31 + 1.12i)T \)
good5 \( 1 + (0.687 + 0.783i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (-0.664 - 0.0874i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (0.920 + 1.86i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (-0.0795 - 0.234i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (1.10 + 1.10i)T + 17iT^{2} \)
19 \( 1 + (-0.117 + 0.588i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (0.440 + 3.34i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-2.11 + 0.138i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (-1.21 + 0.702i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.781 - 3.92i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (0.490 + 3.72i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (-0.983 + 0.484i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (-3.98 + 1.06i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.19 - 6.28i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (3.76 + 4.29i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.0565 - 0.863i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (-10.4 - 5.15i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-0.526 - 1.27i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.849 + 2.05i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.50 + 5.63i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-0.0243 - 0.0213i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (4.99 - 2.06i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (12.8 + 7.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.31554029825713763775341688905, −9.352707086584753581129344487326, −8.503834759341493964579743269276, −8.110106800619089605897890440796, −7.12448891818260341434414051885, −6.21768028811302200257508930297, −4.45170689461586837256584121754, −3.21771166257722289206234157726, −2.20209612488722497677696127154, −0.74627919068189914930624927611, 1.85781949389819067161920747439, 3.14052677510750165166103451212, 4.51571858701789943319493024915, 5.55828424944894122587596481024, 6.88904689240960125185964926251, 7.70822163363900859569367879594, 8.340773105436869095726421131202, 9.328529168288590632617519459539, 9.934602956710713289001919493271, 10.82426390080593881724118881980

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