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Properties

Label 2-288-36.23-c1-0-0
Degree $2$
Conductor $288$
Sign $-0.394 - 0.918i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
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  + (−1.20 − 1.24i)3-s + (−0.398 − 0.229i)5-s + (−4.28 + 2.47i)7-s + (−0.0927 + 2.99i)9-s + (1.17 + 2.03i)11-s + (−0.0384 + 0.0666i)13-s + (0.194 + 0.772i)15-s + 5.92i·17-s − 3.59i·19-s + (8.23 + 2.34i)21-s + (−2.41 + 4.18i)23-s + (−2.39 − 4.14i)25-s + (3.84 − 3.49i)27-s + (−6.54 + 3.78i)29-s + (−0.663 − 0.383i)31-s + ⋯
L(s)  = 1  + (−0.696 − 0.717i)3-s + (−0.178 − 0.102i)5-s + (−1.61 + 0.934i)7-s + (−0.0309 + 0.999i)9-s + (0.354 + 0.613i)11-s + (−0.0106 + 0.0184i)13-s + (0.0501 + 0.199i)15-s + 1.43i·17-s − 0.824i·19-s + (1.79 + 0.511i)21-s + (−0.503 + 0.872i)23-s + (−0.478 − 0.829i)25-s + (0.739 − 0.673i)27-s + (−1.21 + 0.702i)29-s + (−0.119 − 0.0687i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.394 - 0.918i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210493 + 0.319403i\)
\(L(\frac12)\) \(\approx\) \(0.210493 + 0.319403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.20 + 1.24i)T \)
good5 \( 1 + (0.398 + 0.229i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (4.28 - 2.47i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.17 - 2.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0384 - 0.0666i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.92iT - 17T^{2} \)
19 \( 1 + 3.59iT - 19T^{2} \)
23 \( 1 + (2.41 - 4.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.54 - 3.78i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.663 + 0.383i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.47T + 37T^{2} \)
41 \( 1 + (0.986 + 0.569i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.79 - 3.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.01 - 5.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.71iT - 53T^{2} \)
59 \( 1 + (-4.15 + 7.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.63 - 4.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.84 - 1.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 7.51T + 73T^{2} \)
79 \( 1 + (-2.52 + 1.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.44 - 7.69i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.71iT - 89T^{2} \)
97 \( 1 + (-3.16 - 5.47i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.28552529625648712626819137310, −11.40163195430344867835847476916, −10.20778198516481132774892385907, −9.341804217559176745225823946514, −8.232304198326920127751740635074, −6.95073584497119189398077816063, −6.27278212764959447403102316974, −5.34274711181909121960164933843, −3.67399035329487618174881035880, −2.06694632574417257968134240712, 0.29703432341142096066709066126, 3.31302618356576447145605628276, 4.03548393204257878481721351925, 5.54777822762915889724861519364, 6.50037746331793141792025113970, 7.35259396276165795151499825709, 9.005333125361104152903796487293, 9.816749374428876668200926363060, 10.44418296800125206129073288119, 11.47037571812923167574744645509

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