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Properties

Label 2-288-72.43-c2-0-11
Degree $2$
Conductor $288$
Sign $0.947 - 0.318i$
Analytic cond. $7.84743$
Root an. cond. $2.80132$
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  + (−1.21 + 2.74i)3-s + (5.15 − 2.97i)5-s + (4.09 + 2.36i)7-s + (−6.04 − 6.66i)9-s + (6.94 − 12.0i)11-s + (4.03 − 2.32i)13-s + (1.89 + 17.7i)15-s + 21.5·17-s − 3.83·19-s + (−11.4 + 8.34i)21-s + (−30.0 + 17.3i)23-s + (5.23 − 9.07i)25-s + (25.6 − 8.47i)27-s + (39.3 + 22.7i)29-s + (31.8 − 18.4i)31-s + ⋯
L(s)  = 1  + (−0.405 + 0.914i)3-s + (1.03 − 0.595i)5-s + (0.584 + 0.337i)7-s + (−0.671 − 0.740i)9-s + (0.631 − 1.09i)11-s + (0.310 − 0.179i)13-s + (0.126 + 1.18i)15-s + 1.26·17-s − 0.201·19-s + (−0.545 + 0.397i)21-s + (−1.30 + 0.753i)23-s + (0.209 − 0.362i)25-s + (0.949 − 0.313i)27-s + (1.35 + 0.784i)29-s + (1.02 − 0.593i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(7.84743\)
Root analytic conductor: \(2.80132\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.947 - 0.318i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.82207 + 0.298106i\)
\(L(\frac12)\) \(\approx\) \(1.82207 + 0.298106i\)
\(L(2)\) 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 \( 1 + (1.21 - 2.74i)T \)
good5 \( 1 + (-5.15 + 2.97i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-4.09 - 2.36i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-6.94 + 12.0i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.03 + 2.32i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 21.5T + 289T^{2} \)
19 \( 1 + 3.83T + 361T^{2} \)
23 \( 1 + (30.0 - 17.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-39.3 - 22.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-31.8 + 18.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 36.0iT - 1.36e3T^{2} \)
41 \( 1 + (10.2 + 17.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (3.50 - 6.07i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-53.1 - 30.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 58.7iT - 2.80e3T^{2} \)
59 \( 1 + (11.0 + 19.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (47.1 + 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (56.9 + 98.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 8.30iT - 5.04e3T^{2} \)
73 \( 1 + 114.T + 5.32e3T^{2} \)
79 \( 1 + (47.5 + 27.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (15.6 - 27.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 47.7T + 7.92e3T^{2} \)
97 \( 1 + (28.7 - 49.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.66845572719968149021077535781, −10.56259909895873844931697964662, −9.794862646650577172783805286590, −8.924370009548690189813813281438, −8.128265939690903186312781890523, −6.13585141981069241850505514490, −5.68877484245800340125620216900, −4.60255806261418355608047029175, −3.21983082618315425498121476914, −1.25683412684002781914467242840, 1.37428794232487365095437264338, 2.47404978457540394971475577303, 4.44305117717360608924402175313, 5.81577552992110183225749498254, 6.55066915113620676139714457947, 7.48619577043348281519146427705, 8.502513532072867882413225844988, 9.993336482701750883875666952385, 10.44428256606446756873072829750, 11.79593590561342942728669278928

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