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Properties

Label 2-288-36.11-c1-0-5
Degree $2$
Conductor $288$
Sign $0.918 + 0.394i$
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

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

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.918 + 0.394i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.918 + 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68331 - 0.345973i\)
\(L(\frac12)\) \(\approx\) \(1.68331 - 0.345973i\)
\(L(\frac{3}{2})\) 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

−11.62986031849904378768725376163, −11.26024011045536913493704888117, −9.501136527680678410515492841868, −8.874362723598195032492503866476, −7.69135800226031625227045477635, −7.33398319290665446593764249497, −5.66304508475124031558842290379, −4.63806718843444778180525212979, −2.86068677661362030832689507928, −1.73390804359559757239340827855, 1.82707595245846912667528198654, 3.66603302522724521713060279882, 4.48062129747437932661814756034, 5.59360295511731081310584037327, 7.36492934942367178895634205088, 8.241276542049715363222330717546, 8.698827207967393163913118302221, 10.39059620129683096672981325120, 10.59427194676860714663520469612, 11.63337469806207937029626584671

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