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Properties

Label 2-288-72.13-c3-0-10
Degree $2$
Conductor $288$
Sign $0.0562 - 0.998i$
Analytic cond. $16.9925$
Root an. cond. $4.12220$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.64 − 2.32i)3-s + (17.5 + 10.1i)5-s + (10.6 + 18.5i)7-s + (16.1 + 21.6i)9-s + (−1.86 + 1.07i)11-s + (−13.7 − 7.94i)13-s + (−58.0 − 88.0i)15-s − 100.·17-s − 2.35i·19-s + (−6.62 − 110. i)21-s + (−74.6 + 129. i)23-s + (143. + 248. i)25-s + (−24.9 − 138. i)27-s + (175. − 101. i)29-s + (−66.8 + 115. i)31-s + ⋯
L(s)  = 1  + (−0.894 − 0.447i)3-s + (1.57 + 0.907i)5-s + (0.577 + 0.999i)7-s + (0.599 + 0.800i)9-s + (−0.0509 + 0.0294i)11-s + (−0.293 − 0.169i)13-s + (−1.00 − 1.51i)15-s − 1.42·17-s − 0.0284i·19-s + (−0.0688 − 1.15i)21-s + (−0.676 + 1.17i)23-s + (1.14 + 1.98i)25-s + (−0.178 − 0.984i)27-s + (1.12 − 0.647i)29-s + (−0.387 + 0.671i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0562 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0562 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.0562 - 0.998i$
Analytic conductor: \(16.9925\)
Root analytic conductor: \(4.12220\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :3/2),\ 0.0562 - 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.601096153\)
\(L(\frac12)\) \(\approx\) \(1.601096153\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.64 + 2.32i)T \)
good5 \( 1 + (-17.5 - 10.1i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-10.6 - 18.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (1.86 - 1.07i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (13.7 + 7.94i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 100.T + 4.91e3T^{2} \)
19 \( 1 + 2.35iT - 6.85e3T^{2} \)
23 \( 1 + (74.6 - 129. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-175. + 101. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (66.8 - 115. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 16.6iT - 5.06e4T^{2} \)
41 \( 1 + (65.4 - 113. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-107. + 62.2i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-139. - 241. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 342. iT - 1.48e5T^{2} \)
59 \( 1 + (191. + 110. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (367. - 212. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-47.7 - 27.5i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 996.T + 3.57e5T^{2} \)
73 \( 1 + 910.T + 3.89e5T^{2} \)
79 \( 1 + (163. + 283. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-147. + 85.4i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 850.T + 7.04e5T^{2} \)
97 \( 1 + (24.9 + 43.1i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.50538451116601273338264272290, −10.73635925099912914986865852654, −9.900432351329996820210870180692, −8.886646372730642632705341767180, −7.46738291108606103738578866757, −6.38827592531431984105269283351, −5.80946762479483368449410657128, −4.85280796726995096659118006903, −2.55523738301418257562246539383, −1.72895581124438597046600274891, 0.66642826745131862806394186559, 1.99386052982953889382728658221, 4.33981064660973930616136274613, 4.93459209488600153040881468833, 6.05598977358620406495424443003, 6.89634069014719982099101153303, 8.517548452783847986396889875552, 9.437312653879847163788170204393, 10.31271754872095796233579816843, 10.87665016130576183256661663519

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