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Properties

Label 2-24e2-144.11-c1-0-3
Degree $2$
Conductor $576$
Sign $-0.195 - 0.980i$
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  + (1.58 + 0.692i)3-s + (−2.83 − 0.759i)5-s + (−1.41 + 2.45i)7-s + (2.04 + 2.19i)9-s + (−0.794 + 0.212i)11-s + (3.22 + 0.864i)13-s + (−3.97 − 3.16i)15-s + 7.28i·17-s + (0.951 + 0.951i)19-s + (−3.94 + 2.91i)21-s + (−5.13 + 2.96i)23-s + (3.12 + 1.80i)25-s + (1.71 + 4.90i)27-s + (−1.76 + 0.473i)29-s + (−2.05 + 1.18i)31-s + ⋯
L(s)  = 1  + (0.916 + 0.399i)3-s + (−1.26 − 0.339i)5-s + (−0.535 + 0.927i)7-s + (0.680 + 0.732i)9-s + (−0.239 + 0.0641i)11-s + (0.894 + 0.239i)13-s + (−1.02 − 0.817i)15-s + 1.76i·17-s + (0.218 + 0.218i)19-s + (−0.861 + 0.636i)21-s + (−1.07 + 0.617i)23-s + (0.624 + 0.360i)25-s + (0.330 + 0.943i)27-s + (−0.328 + 0.0879i)29-s + (−0.369 + 0.213i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.195 - 0.980i$
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: \((2,\ 576,\ (\ :1/2),\ -0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.826002 + 1.00714i\)
\(L(\frac12)\) \(\approx\) \(0.826002 + 1.00714i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 0.692i)T \)
good5 \( 1 + (2.83 + 0.759i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.41 - 2.45i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.794 - 0.212i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.22 - 0.864i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 7.28iT - 17T^{2} \)
19 \( 1 + (-0.951 - 0.951i)T + 19iT^{2} \)
23 \( 1 + (5.13 - 2.96i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.76 - 0.473i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (2.05 - 1.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.03 - 6.03i)T + 37iT^{2} \)
41 \( 1 + (4.60 + 7.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.402 + 1.50i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.85 + 4.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.50 + 4.50i)T - 53iT^{2} \)
59 \( 1 + (-2.85 + 10.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.16 + 8.08i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.99 + 11.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.98iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (-8.94 - 5.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.00 - 3.74i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 9.25T + 89T^{2} \)
97 \( 1 + (-0.148 + 0.257i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.92251114366811674958277476748, −9.987734048301288038579641909082, −9.024190715733299165972367981127, −8.295988963409994370051232566321, −7.86118033641489866044267409446, −6.49165119740051382857950793312, −5.32890677693874207444527845154, −3.90975307290840928472403555192, −3.56933980683425063697683839931, −1.98105843868406737901070526360, 0.67199763112494267187218038478, 2.74523690389581908343348697576, 3.63831037021551105472571280689, 4.39991672553733797849243094713, 6.19228615350761550072823657989, 7.36654123864706213099572372506, 7.52571123615091253288703397685, 8.582774156737338427450033255086, 9.529121151041146887324394287445, 10.45708094061340353809656960580

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