Properties

Label 2-24e2-576.205-c1-0-56
Degree $2$
Conductor $576$
Sign $-0.488 + 0.872i$
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  + (−0.836 + 1.14i)2-s + (−1.64 − 0.535i)3-s + (−0.600 − 1.90i)4-s + (−2.22 − 2.53i)5-s + (1.98 − 1.43i)6-s + (3.36 + 0.442i)7-s + (2.67 + 0.911i)8-s + (2.42 + 1.76i)9-s + (4.74 − 0.413i)10-s + (−0.977 − 1.98i)11-s + (−0.0325 + 3.46i)12-s + (0.602 + 1.77i)13-s + (−3.31 + 3.46i)14-s + (2.30 + 5.35i)15-s + (−3.27 + 2.29i)16-s + (−1.12 − 1.12i)17-s + ⋯
L(s)  = 1  + (−0.591 + 0.806i)2-s + (−0.951 − 0.309i)3-s + (−0.300 − 0.953i)4-s + (−0.993 − 1.13i)5-s + (0.811 − 0.583i)6-s + (1.27 + 0.167i)7-s + (0.946 + 0.322i)8-s + (0.808 + 0.587i)9-s + (1.50 − 0.130i)10-s + (−0.294 − 0.597i)11-s + (−0.00940 + 0.999i)12-s + (0.167 + 0.492i)13-s + (−0.886 + 0.925i)14-s + (0.594 + 1.38i)15-s + (−0.819 + 0.572i)16-s + (−0.272 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.488 + 0.872i$
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: \((2,\ 576,\ (\ :1/2),\ -0.488 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200854 - 0.342595i\)
\(L(\frac12)\) \(\approx\) \(0.200854 - 0.342595i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.836 - 1.14i)T \)
3 \( 1 + (1.64 + 0.535i)T \)
good5 \( 1 + (2.22 + 2.53i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (-3.36 - 0.442i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (0.977 + 1.98i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (-0.602 - 1.77i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (1.12 + 1.12i)T + 17iT^{2} \)
19 \( 1 + (-0.386 + 1.94i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (0.921 + 6.99i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-3.27 + 0.214i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (6.32 - 3.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.66 + 8.35i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.295 - 2.24i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (3.30 - 1.63i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (5.09 - 1.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.28 - 7.91i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (9.67 + 11.0i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.655 - 9.99i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (4.60 + 2.26i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (-3.13 - 7.57i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-4.74 + 11.4i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.01 + 14.9i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (1.61 + 1.41i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (9.11 - 3.77i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-6.11 - 3.53i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.72010773143834316443655473498, −9.212450332874262960174567326545, −8.435822924311749219996467200827, −7.87861629801985578346233674202, −6.96234335017263236610600558649, −5.83062376636174219243847324949, −4.82679919059302826055775444678, −4.49661809562383928134587454450, −1.59685578815251482400115022260, −0.33281807617784433871563257264, 1.64115659839555917812399686902, 3.34494725694858818054925431972, 4.21848633283013764829974440549, 5.23794904794577337584799651024, 6.80074735676281428832521961465, 7.62655232201352863869775684311, 8.186939904870497981385945300717, 9.659176666287471153020480459702, 10.43832397684267932176536241836, 11.07165193647613408072984099099

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