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Properties

Label 2-12e2-48.29-c2-0-11
Degree $2$
Conductor $144$
Sign $-0.974 + 0.223i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.56 − 1.24i)2-s + (0.888 + 3.90i)4-s + (−1.84 − 1.84i)5-s − 0.226i·7-s + (3.47 − 7.20i)8-s + (0.583 + 5.18i)10-s + (−11.5 − 11.5i)11-s + (−8.11 − 8.11i)13-s + (−0.282 + 0.354i)14-s + (−14.4 + 6.92i)16-s + 6.20i·17-s + (−4.43 − 4.43i)19-s + (5.55 − 8.83i)20-s + (3.66 + 32.5i)22-s − 28.6·23-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)2-s + (0.222 + 0.975i)4-s + (−0.369 − 0.369i)5-s − 0.0323i·7-s + (0.434 − 0.900i)8-s + (0.0583 + 0.518i)10-s + (−1.05 − 1.05i)11-s + (−0.624 − 0.624i)13-s + (−0.0201 + 0.0253i)14-s + (−0.901 + 0.433i)16-s + 0.364i·17-s + (−0.233 − 0.233i)19-s + (0.277 − 0.441i)20-s + (0.166 + 1.47i)22-s − 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.974 + 0.223i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.974 + 0.223i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0493098 - 0.436174i\)
\(L(\frac12)\) \(\approx\) \(0.0493098 - 0.436174i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.56 + 1.24i)T \)
3 \( 1 \)
good5 \( 1 + (1.84 + 1.84i)T + 25iT^{2} \)
7 \( 1 + 0.226iT - 49T^{2} \)
11 \( 1 + (11.5 + 11.5i)T + 121iT^{2} \)
13 \( 1 + (8.11 + 8.11i)T + 169iT^{2} \)
17 \( 1 - 6.20iT - 289T^{2} \)
19 \( 1 + (4.43 + 4.43i)T + 361iT^{2} \)
23 \( 1 + 28.6T + 529T^{2} \)
29 \( 1 + (15.7 - 15.7i)T - 841iT^{2} \)
31 \( 1 + 33.5T + 961T^{2} \)
37 \( 1 + (-43.8 + 43.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 62.2T + 1.68e3T^{2} \)
43 \( 1 + (-18.3 + 18.3i)T - 1.84e3iT^{2} \)
47 \( 1 - 13.8iT - 2.20e3T^{2} \)
53 \( 1 + (-37.9 - 37.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (70.4 + 70.4i)T + 3.48e3iT^{2} \)
61 \( 1 + (60.3 + 60.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (-61.9 - 61.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 32.5T + 5.04e3T^{2} \)
73 \( 1 - 130. iT - 5.32e3T^{2} \)
79 \( 1 + 132.T + 6.24e3T^{2} \)
83 \( 1 + (-19.2 + 19.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 91.2T + 7.92e3T^{2} \)
97 \( 1 + 20.0T + 9.40e3T^{2} \)
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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

−12.41993534764403838605366173104, −11.13772361524814141910273374513, −10.49226034786487459865036241158, −9.288605162275451583950459411104, −8.206887366717272595568103098360, −7.53734507437421449204549443458, −5.74891706782081161781845618762, −4.03553343222891649819162479562, −2.54443329238596976337103372210, −0.34767972529487381278156560608, 2.24054879612610557346695005570, 4.53633650543697621963025371872, 5.89094854286924666856472488122, 7.29767733716531807378785062474, 7.77656253029840488186374231248, 9.263301425745069677396514031280, 10.07056823491612544328635157800, 11.08224374990880413778933563402, 12.17104130315446387673186213734, 13.49221992909211159641342897267

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