Properties

Label 2-288-72.5-c4-0-30
Degree $2$
Conductor $288$
Sign $0.621 + 0.783i$
Analytic cond. $29.7705$
Root an. cond. $5.45623$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.20 + 5.39i)3-s + (21.1 − 36.5i)5-s + (16.9 + 29.2i)7-s + (22.8 − 77.7i)9-s + (28.4 + 49.2i)11-s + (79.2 + 45.7i)13-s + (45.1 + 377. i)15-s + 122. i·17-s − 551. i·19-s + (−279. − 119. i)21-s + (−595. − 344. i)23-s + (−578. − 1.00e3i)25-s + (254. + 683. i)27-s + (665. + 1.15e3i)29-s + (−305. + 529. i)31-s + ⋯
L(s)  = 1  + (−0.800 + 0.599i)3-s + (0.844 − 1.46i)5-s + (0.344 + 0.597i)7-s + (0.281 − 0.959i)9-s + (0.234 + 0.406i)11-s + (0.469 + 0.270i)13-s + (0.200 + 1.67i)15-s + 0.425i·17-s − 1.52i·19-s + (−0.634 − 0.271i)21-s + (−1.12 − 0.650i)23-s + (−0.926 − 1.60i)25-s + (0.349 + 0.936i)27-s + (0.791 + 1.37i)29-s + (−0.317 + 0.550i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.621 + 0.783i$
Analytic conductor: \(29.7705\)
Root analytic conductor: \(5.45623\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :2),\ 0.621 + 0.783i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.708490324\)
\(L(\frac12)\) \(\approx\) \(1.708490324\)
\(L(3)\) 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 \( 1 + (7.20 - 5.39i)T \)
good5 \( 1 + (-21.1 + 36.5i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (-16.9 - 29.2i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-28.4 - 49.2i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-79.2 - 45.7i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 122. iT - 8.35e4T^{2} \)
19 \( 1 + 551. iT - 1.30e5T^{2} \)
23 \( 1 + (595. + 344. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-665. - 1.15e3i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (305. - 529. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.55e3iT - 1.87e6T^{2} \)
41 \( 1 + (978. + 564. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-2.63e3 + 1.52e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-1.41e3 + 815. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 2.78e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.22e3 + 2.11e3i)T + (-6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-4.58e3 + 2.64e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.53e3 + 2.04e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 1.18e3iT - 2.54e7T^{2} \)
73 \( 1 - 585.T + 2.83e7T^{2} \)
79 \( 1 + (-1.96e3 - 3.39e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (5.66e3 + 9.81e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 6.30e3iT - 6.27e7T^{2} \)
97 \( 1 + (-373. - 646. i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−10.99590556843417444959848317485, −10.06414475999483916233199596809, −8.963170159471700054482874314092, −8.736954178565786821977981146723, −6.82808238142048515453484021786, −5.70291715149871154722765841379, −5.05338700937309800811288704648, −4.12678864153131895010456219324, −1.98494076429360433797557681845, −0.67169238810916981557961070618, 1.20860445490803293021579328406, 2.49254870622576427184776040620, 4.00328877294247703761333232938, 5.78657799884186949212191054068, 6.18037267764315664932337686172, 7.28284373553181081734388848355, 8.063579151755749252016540986082, 9.909634766728420457653531148399, 10.38504614030665632629306149193, 11.28403648948796313001214323045

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