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Properties

Label 2-288-32.5-c1-0-4
Degree $2$
Conductor $288$
Sign $-0.212 - 0.977i$
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  + (0.835 + 1.14i)2-s + (−0.604 + 1.90i)4-s + (0.823 + 0.341i)5-s + (0.760 + 0.760i)7-s + (−2.68 + 0.903i)8-s + (0.298 + 1.22i)10-s + (−1.75 + 4.24i)11-s + (2.78 − 1.15i)13-s + (−0.232 + 1.50i)14-s + (−3.26 − 2.30i)16-s + 0.00932i·17-s + (5.33 − 2.21i)19-s + (−1.14 + 1.36i)20-s + (−6.31 + 1.54i)22-s + (−2.06 + 2.06i)23-s + ⋯
L(s)  = 1  + (0.590 + 0.806i)2-s + (−0.302 + 0.953i)4-s + (0.368 + 0.152i)5-s + (0.287 + 0.287i)7-s + (−0.947 + 0.319i)8-s + (0.0944 + 0.387i)10-s + (−0.530 + 1.28i)11-s + (0.773 − 0.320i)13-s + (−0.0621 + 0.401i)14-s + (−0.817 − 0.575i)16-s + 0.00226i·17-s + (1.22 − 0.507i)19-s + (−0.256 + 0.305i)20-s + (−1.34 + 0.328i)22-s + (−0.430 + 0.430i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.212 - 0.977i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.212 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09054 + 1.35378i\)
\(L(\frac12)\) \(\approx\) \(1.09054 + 1.35378i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.835 - 1.14i)T \)
3 \( 1 \)
good5 \( 1 + (-0.823 - 0.341i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.760 - 0.760i)T + 7iT^{2} \)
11 \( 1 + (1.75 - 4.24i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.78 + 1.15i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.00932iT - 17T^{2} \)
19 \( 1 + (-5.33 + 2.21i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.06 - 2.06i)T - 23iT^{2} \)
29 \( 1 + (1.24 + 2.99i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 + (-4.94 - 2.04i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-7.21 + 7.21i)T - 41iT^{2} \)
43 \( 1 + (-2.68 + 6.49i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 + (-4.21 + 10.1i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.98 + 2.06i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-4.20 - 10.1i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (1.34 + 3.23i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.86 + 5.86i)T + 71iT^{2} \)
73 \( 1 + (1.26 - 1.26i)T - 73iT^{2} \)
79 \( 1 - 13.6iT - 79T^{2} \)
83 \( 1 + (-6.49 + 2.69i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (10.7 + 10.7i)T + 89iT^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−12.25054139441790186133069978056, −11.38876631128095975115037620498, −10.05159602133178222845938245316, −9.130700366875050908896126200358, −7.939301476040601954640064482093, −7.22814500966120240818566551379, −5.98699865279194590285188504528, −5.17818711480300947488088369773, −3.96640127859181809814277021058, −2.43177265465671022184911926584, 1.27658826632230467080868892598, 2.96660881353489350727768300441, 4.10531669680177765968432008414, 5.46842489768645327985566055719, 6.12388487328814131368087672561, 7.76553065118794721556075054155, 8.936236681516433837249703469221, 9.823617418945391605306979616362, 10.92734896318105580797813860942, 11.36020009768545097927248940341

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