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Properties

Label 2-288-96.83-c1-0-2
Degree $2$
Conductor $288$
Sign $0.597 - 0.801i$
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  + (−1.40 + 0.144i)2-s + (1.95 − 0.405i)4-s + (0.366 + 0.885i)5-s + (1.21 + 1.21i)7-s + (−2.69 + 0.853i)8-s + (−0.643 − 1.19i)10-s + (0.545 + 1.31i)11-s + (0.0270 + 0.0112i)13-s + (−1.88 − 1.53i)14-s + (3.67 − 1.58i)16-s + 1.32·17-s + (−1.73 + 4.19i)19-s + (1.07 + 1.58i)20-s + (−0.956 − 1.77i)22-s + (0.934 + 0.934i)23-s + ⋯
L(s)  = 1  + (−0.994 + 0.101i)2-s + (0.979 − 0.202i)4-s + (0.164 + 0.396i)5-s + (0.459 + 0.459i)7-s + (−0.953 + 0.301i)8-s + (−0.203 − 0.377i)10-s + (0.164 + 0.396i)11-s + (0.00750 + 0.00310i)13-s + (−0.503 − 0.409i)14-s + (0.917 − 0.397i)16-s + 0.321·17-s + (−0.398 + 0.961i)19-s + (0.241 + 0.354i)20-s + (−0.204 − 0.378i)22-s + (0.194 + 0.194i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.597 - 0.801i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.597 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.800480 + 0.401503i\)
\(L(\frac12)\) \(\approx\) \(0.800480 + 0.401503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.144i)T \)
3 \( 1 \)
good5 \( 1 + (-0.366 - 0.885i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.21 - 1.21i)T + 7iT^{2} \)
11 \( 1 + (-0.545 - 1.31i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.0270 - 0.0112i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 1.32T + 17T^{2} \)
19 \( 1 + (1.73 - 4.19i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.934 - 0.934i)T + 23iT^{2} \)
29 \( 1 + (-9.35 - 3.87i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 9.74iT - 31T^{2} \)
37 \( 1 + (6.28 - 2.60i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.42 + 3.42i)T - 41iT^{2} \)
43 \( 1 + (-0.997 + 0.413i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 6.21iT - 47T^{2} \)
53 \( 1 + (-2.94 + 1.22i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (10.4 - 4.32i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.76 + 6.68i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (10.1 + 4.18i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-7.38 + 7.38i)T - 71iT^{2} \)
73 \( 1 + (8.30 + 8.30i)T + 73iT^{2} \)
79 \( 1 - 5.54T + 79T^{2} \)
83 \( 1 + (11.4 + 4.74i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (7.93 + 7.93i)T + 89iT^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−11.97034891569004050345582529453, −10.60051284609201518007387093723, −10.26521549094896200672246981844, −8.950887535951269057371495570411, −8.310907068068739112928721563347, −7.14861487983914944215320220701, −6.29895430963666210641540019688, −5.04182068353341926188830942204, −3.10812528686199959655241048674, −1.68138469099534983003016918868, 1.02832448619123131900019738985, 2.71880782765492099747307181518, 4.39827597367892383998889192625, 5.88388058304493826432904321676, 6.99723368943300524484056342476, 7.995625844298617697613338855826, 8.830437866271200713075107885407, 9.702947719706991430697925062121, 10.73457995430535319454554420304, 11.38920057136265341805706491969

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