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Properties

Label 2-288-96.59-c1-0-0
Degree $2$
Conductor $288$
Sign $-0.993 + 0.113i$
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  + (−1.04 + 0.948i)2-s + (0.201 − 1.98i)4-s + (−1.39 + 3.36i)5-s + (−1.05 + 1.05i)7-s + (1.67 + 2.27i)8-s + (−1.73 − 4.85i)10-s + (1.50 − 3.64i)11-s + (−2.23 + 0.927i)13-s + (0.106 − 2.11i)14-s + (−3.91 − 0.803i)16-s − 7.49·17-s + (0.818 + 1.97i)19-s + (6.42 + 3.45i)20-s + (1.87 + 5.25i)22-s + (−5.80 + 5.80i)23-s + ⋯
L(s)  = 1  + (−0.741 + 0.670i)2-s + (0.100 − 0.994i)4-s + (−0.624 + 1.50i)5-s + (−0.399 + 0.399i)7-s + (0.592 + 0.805i)8-s + (−0.547 − 1.53i)10-s + (0.455 − 1.09i)11-s + (−0.620 + 0.257i)13-s + (0.0285 − 0.564i)14-s + (−0.979 − 0.200i)16-s − 1.81·17-s + (0.187 + 0.453i)19-s + (1.43 + 0.773i)20-s + (0.399 + 1.12i)22-s + (−1.21 + 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.993 + 0.113i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.993 + 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0231172 - 0.407104i\)
\(L(\frac12)\) \(\approx\) \(0.0231172 - 0.407104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 - 0.948i)T \)
3 \( 1 \)
good5 \( 1 + (1.39 - 3.36i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.05 - 1.05i)T - 7iT^{2} \)
11 \( 1 + (-1.50 + 3.64i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.23 - 0.927i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 7.49T + 17T^{2} \)
19 \( 1 + (-0.818 - 1.97i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.80 - 5.80i)T - 23iT^{2} \)
29 \( 1 + (-0.326 + 0.135i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.71iT - 31T^{2} \)
37 \( 1 + (-0.387 - 0.160i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.50 + 1.50i)T + 41iT^{2} \)
43 \( 1 + (-7.40 - 3.06i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.27iT - 47T^{2} \)
53 \( 1 + (-3.94 - 1.63i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-12.7 - 5.30i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (0.579 + 1.40i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (7.96 - 3.29i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-4.75 - 4.75i)T + 71iT^{2} \)
73 \( 1 + (7.99 - 7.99i)T - 73iT^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (-1.35 + 0.560i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.75 - 4.75i)T - 89iT^{2} \)
97 \( 1 + 1.28T + 97T^{2} \)
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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

−11.82853008858847264252153927854, −11.22439661132890816305742472385, −10.38774956768825500754428992370, −9.389287230298401641675438830838, −8.413226016220362135152031669581, −7.34941176838530645909015483455, −6.60437551599546532933330473984, −5.77832410401943335085036197937, −3.93515108898715314520523499884, −2.45616333469912965613660652885, 0.36938717273129843383709435045, 2.15638409908895952189396073414, 4.10869651084929825397961580184, 4.65597494930544981717037095160, 6.71230112975096334371059027593, 7.71030554538504806404338262710, 8.682337653909842325463744081104, 9.355952290736937928038429639450, 10.25282645752277394823964608952, 11.41745230474379093832479487904

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