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Properties

Label 2-12e2-48.5-c2-0-14
Degree $2$
Conductor $144$
Sign $0.114 + 0.993i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.114 + 0.993i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.114 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.67662 - 1.49457i\)
\(L(\frac12)\) \(\approx\) \(1.67662 - 1.49457i\)
\(L(2)\) 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

−12.66614211672316699713098778801, −11.67533205286429790830303667208, −10.88384509776743987363871082757, −9.596794332700309601093568941104, −8.796692294652619510355663417902, −6.88897345763697748700311750042, −5.82702853530184315233021922202, −4.61438746581803124885639942186, −3.24157171609234626664587008988, −1.42852451231928204170509825048, 2.52806253365658849013488857646, 4.14092474050590394171501853869, 5.35203437610641816169819596012, 6.67265143545649160479963857641, 7.38264643431081863073055998576, 8.830752427051509270743552751327, 9.986276567408966421235866415721, 11.33832773099516475049734165684, 12.34371277645577359674193434151, 13.12145238774784749192884068020

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