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Properties

Label 2-12e2-16.5-c3-0-6
Degree $2$
Conductor $144$
Sign $-0.0551 - 0.998i$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 + 0.316i)2-s + (7.80 − 1.77i)4-s + (12.6 + 12.6i)5-s + 13.8i·7-s + (−21.3 + 7.45i)8-s + (−39.5 − 31.5i)10-s + (−1.54 − 1.54i)11-s + (32.7 − 32.7i)13-s + (−4.38 − 38.9i)14-s + (57.6 − 27.7i)16-s − 18.6·17-s + (−86.4 + 86.4i)19-s + (121. + 76.1i)20-s + (4.83 + 3.85i)22-s + 134. i·23-s + ⋯
L(s)  = 1  + (−0.993 + 0.111i)2-s + (0.975 − 0.222i)4-s + (1.13 + 1.13i)5-s + 0.749i·7-s + (−0.944 + 0.329i)8-s + (−1.25 − 0.997i)10-s + (−0.0424 − 0.0424i)11-s + (0.699 − 0.699i)13-s + (−0.0837 − 0.744i)14-s + (0.901 − 0.433i)16-s − 0.266·17-s + (−1.04 + 1.04i)19-s + (1.35 + 0.851i)20-s + (0.0468 + 0.0373i)22-s + 1.21i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.0551 - 0.998i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ -0.0551 - 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.818939 + 0.865458i\)
\(L(\frac12)\) \(\approx\) \(0.818939 + 0.865458i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.81 - 0.316i)T \)
3 \( 1 \)
good5 \( 1 + (-12.6 - 12.6i)T + 125iT^{2} \)
7 \( 1 - 13.8iT - 343T^{2} \)
11 \( 1 + (1.54 + 1.54i)T + 1.33e3iT^{2} \)
13 \( 1 + (-32.7 + 32.7i)T - 2.19e3iT^{2} \)
17 \( 1 + 18.6T + 4.91e3T^{2} \)
19 \( 1 + (86.4 - 86.4i)T - 6.85e3iT^{2} \)
23 \( 1 - 134. iT - 1.21e4T^{2} \)
29 \( 1 + (-59.7 + 59.7i)T - 2.43e4iT^{2} \)
31 \( 1 + 31.5T + 2.97e4T^{2} \)
37 \( 1 + (-89.1 - 89.1i)T + 5.06e4iT^{2} \)
41 \( 1 - 210. iT - 6.89e4T^{2} \)
43 \( 1 + (-119. - 119. i)T + 7.95e4iT^{2} \)
47 \( 1 - 182.T + 1.03e5T^{2} \)
53 \( 1 + (-26.1 - 26.1i)T + 1.48e5iT^{2} \)
59 \( 1 + (441. + 441. i)T + 2.05e5iT^{2} \)
61 \( 1 + (174. - 174. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-91.7 + 91.7i)T - 3.00e5iT^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 + 299. iT - 3.89e5T^{2} \)
79 \( 1 + 943.T + 4.93e5T^{2} \)
83 \( 1 + (313. - 313. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−12.84541343855384294448684690501, −11.50575607045449561624100838509, −10.59455447313944548215834282698, −9.885339809484636447754662950104, −8.842507290857443519395162189822, −7.68943237830934443516754420151, −6.29279555739323151173909971213, −5.82036350669078123471715467839, −3.03770291268963847978446930183, −1.83318409347529306806793790367, 0.814880648808836010121158582553, 2.17805879514106895722939702890, 4.39962444975856434440545189954, 6.01460871263218620197625963355, 7.03894548893802357262282191250, 8.637706538763130638523683114029, 9.050833517577840020311002493963, 10.24230792451018795961847331661, 11.01249080812167560604755567479, 12.40660507603204769831369288901

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