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Properties

Label 2-288-32.29-c1-0-18
Degree $2$
Conductor $288$
Sign $-0.799 - 0.600i$
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  + (0.126 − 1.40i)2-s + (−1.96 − 0.355i)4-s + (−1.36 − 3.28i)5-s + (−2.73 + 2.73i)7-s + (−0.749 + 2.72i)8-s + (−4.80 + 1.50i)10-s + (−3.01 + 1.24i)11-s + (0.932 − 2.25i)13-s + (3.50 + 4.19i)14-s + (3.74 + 1.40i)16-s − 0.517i·17-s + (1.52 − 3.68i)19-s + (1.51 + 6.95i)20-s + (1.37 + 4.40i)22-s + (−2.39 − 2.39i)23-s + ⋯
L(s)  = 1  + (0.0893 − 0.996i)2-s + (−0.984 − 0.177i)4-s + (−0.609 − 1.47i)5-s + (−1.03 + 1.03i)7-s + (−0.265 + 0.964i)8-s + (−1.51 + 0.475i)10-s + (−0.909 + 0.376i)11-s + (0.258 − 0.624i)13-s + (0.935 + 1.12i)14-s + (0.936 + 0.350i)16-s − 0.125i·17-s + (0.350 − 0.845i)19-s + (0.337 + 1.55i)20-s + (0.293 + 0.939i)22-s + (−0.500 − 0.500i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.799 - 0.600i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.799 - 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140184 + 0.420520i\)
\(L(\frac12)\) \(\approx\) \(0.140184 + 0.420520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.126 + 1.40i)T \)
3 \( 1 \)
good5 \( 1 + (1.36 + 3.28i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.73 - 2.73i)T - 7iT^{2} \)
11 \( 1 + (3.01 - 1.24i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.932 + 2.25i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.517iT - 17T^{2} \)
19 \( 1 + (-1.52 + 3.68i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.39 + 2.39i)T + 23iT^{2} \)
29 \( 1 + (7.09 + 2.93i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.50T + 31T^{2} \)
37 \( 1 + (3.40 + 8.22i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.21 - 3.21i)T + 41iT^{2} \)
43 \( 1 + (1.31 - 0.544i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.67iT - 47T^{2} \)
53 \( 1 + (-4.19 + 1.73i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.680 - 1.64i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-6.71 - 2.78i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-11.1 - 4.61i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (1.86 - 1.86i)T - 71iT^{2} \)
73 \( 1 + (9.06 + 9.06i)T + 73iT^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + (-1.89 + 4.57i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.70 + 2.70i)T - 89iT^{2} \)
97 \( 1 - 3.73T + 97T^{2} \)
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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

−11.51683545271113288137728120096, −10.27267468952400554256440642079, −9.286286388471656453325009554963, −8.750483813550488049446082573140, −7.71958830016952854809583442918, −5.73079973009329952367023999936, −5.01431537172867745557716424808, −3.74560484799198107598567662308, −2.39248606671852407551784442437, −0.31323637265258204152180984809, 3.29514351247252158478379139936, 3.93160037696567779355231315490, 5.69942629946339270585924206827, 6.74038942167585712056043299410, 7.29820997005852552478498939020, 8.160302764607802659209660495379, 9.653982039483864840189974293657, 10.36271196237036721057184260256, 11.27192118144059937899062885060, 12.61459524009004547792809011339

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