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Properties

Label 2-288-96.11-c1-0-14
Degree $2$
Conductor $288$
Sign $0.447 + 0.894i$
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  + (1.39 − 0.248i)2-s + (1.87 − 0.691i)4-s + (−4.01 − 1.66i)5-s + (2.72 − 2.72i)7-s + (2.44 − 1.42i)8-s + (−6.00 − 1.31i)10-s + (2.48 + 1.02i)11-s + (−0.146 − 0.352i)13-s + (3.11 − 4.46i)14-s + (3.04 − 2.59i)16-s + 1.69·17-s + (−3.86 + 1.60i)19-s + (−8.69 − 0.344i)20-s + (3.70 + 0.814i)22-s + (−3.96 + 3.96i)23-s + ⋯
L(s)  = 1  + (0.984 − 0.175i)2-s + (0.938 − 0.345i)4-s + (−1.79 − 0.744i)5-s + (1.02 − 1.02i)7-s + (0.863 − 0.505i)8-s + (−1.90 − 0.417i)10-s + (0.748 + 0.309i)11-s + (−0.0405 − 0.0978i)13-s + (0.832 − 1.19i)14-s + (0.760 − 0.648i)16-s + 0.411·17-s + (−0.887 + 0.367i)19-s + (−1.94 − 0.0771i)20-s + (0.790 + 0.173i)22-s + (−0.827 + 0.827i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67538 - 1.03563i\)
\(L(\frac12)\) \(\approx\) \(1.67538 - 1.03563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.248i)T \)
3 \( 1 \)
good5 \( 1 + (4.01 + 1.66i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.72 + 2.72i)T - 7iT^{2} \)
11 \( 1 + (-2.48 - 1.02i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.146 + 0.352i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 1.69T + 17T^{2} \)
19 \( 1 + (3.86 - 1.60i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.96 - 3.96i)T - 23iT^{2} \)
29 \( 1 + (-0.582 - 1.40i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 0.247iT - 31T^{2} \)
37 \( 1 + (2.88 - 6.95i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-4.26 - 4.26i)T + 41iT^{2} \)
43 \( 1 + (1.21 - 2.92i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 8.76iT - 47T^{2} \)
53 \( 1 + (-3.22 + 7.78i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.77 - 4.28i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-6.31 + 2.61i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-0.346 - 0.837i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-9.14 - 9.14i)T + 71iT^{2} \)
73 \( 1 + (-0.0835 + 0.0835i)T - 73iT^{2} \)
79 \( 1 + 9.01T + 79T^{2} \)
83 \( 1 + (2.10 + 5.07i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (3.77 - 3.77i)T - 89iT^{2} \)
97 \( 1 + 2.20T + 97T^{2} \)
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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

−11.69266411672174542660812991726, −11.20653954048453497349710502000, −10.09835451740521736557593679921, −8.380916195283145656986955279688, −7.71311400349369452393961221803, −6.84092654954521863809246439008, −5.12614963797307933091149095219, −4.23018574664095126991940285447, −3.70223594573661968784585692820, −1.31809357442747297437117499845, 2.46597807388585813868166037137, 3.80481908735748344165263271981, 4.58496640319607207759764356480, 5.98078225730268286035405676144, 7.05716844001800950557627837317, 7.988317065443013982218829313430, 8.670726307437153412456871969669, 10.74619519762106994730199914110, 11.27177136883581991756410809473, 12.08773131652585734222063947816

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