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Properties

Label 2-24e2-9.5-c2-0-23
Degree $2$
Conductor $576$
Sign $0.234 + 0.972i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.686 + 2.92i)3-s + (−6.55 − 3.78i)5-s + (4.55 + 7.89i)7-s + (−8.05 − 4.00i)9-s + (−0.383 + 0.221i)11-s + (−5.55 + 9.62i)13-s + (15.5 − 16.5i)15-s − 8.01i·17-s − 8.11·19-s + (−26.1 + 7.89i)21-s + (−20.4 − 11.8i)23-s + (16.1 + 28.0i)25-s + (17.2 − 20.7i)27-s + (45.9 − 26.5i)29-s + (14.6 − 25.4i)31-s + ⋯
L(s)  = 1  + (−0.228 + 0.973i)3-s + (−1.31 − 0.757i)5-s + (0.651 + 1.12i)7-s + (−0.895 − 0.445i)9-s + (−0.0348 + 0.0201i)11-s + (−0.427 + 0.740i)13-s + (1.03 − 1.10i)15-s − 0.471i·17-s − 0.427·19-s + (−1.24 + 0.375i)21-s + (−0.888 − 0.513i)23-s + (0.647 + 1.12i)25-s + (0.638 − 0.769i)27-s + (1.58 − 0.913i)29-s + (0.473 − 0.819i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.234 + 0.972i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.234 + 0.972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5050657867\)
\(L(\frac12)\) \(\approx\) \(0.5050657867\)
\(L(2)\) 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 \( 1 + (0.686 - 2.92i)T \)
good5 \( 1 + (6.55 + 3.78i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.55 - 7.89i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.383 - 0.221i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.55 - 9.62i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 8.01iT - 289T^{2} \)
19 \( 1 + 8.11T + 361T^{2} \)
23 \( 1 + (20.4 + 11.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-45.9 + 26.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-14.6 + 25.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 18.4T + 1.36e3T^{2} \)
41 \( 1 + (38.9 + 22.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (11.5 + 19.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-7.32 + 4.22i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 60.5iT - 2.80e3T^{2} \)
59 \( 1 + (-65.9 - 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-2.67 - 4.63i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.8 + 95.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 16.0iT - 5.04e3T^{2} \)
73 \( 1 + 4.35T + 5.32e3T^{2} \)
79 \( 1 + (-0.792 - 1.37i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (7.32 - 4.22i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 64.1iT - 7.92e3T^{2} \)
97 \( 1 + (57.6 + 99.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.32237675253564246757470624329, −9.362765809278932739723777683496, −8.459354490252331849527759513841, −8.148785018295933471387084680797, −6.62277375253424757862300992004, −5.36010100822824520991443427932, −4.63724361381023645291200817643, −3.92107054738834698833586777619, −2.40799943848669390954191755520, −0.22276757648975246295776749337, 1.18675262132833167762296083529, 2.87577714873654732070445854484, 3.96608926035388466838536266855, 5.10077246551337794307658728208, 6.55846744460646930940636159180, 7.16339103816803515449075894351, 7.984552884283637606343944062469, 8.357905808808089418759312331421, 10.30966942164301887687609879736, 10.72657193068574839327521959883

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