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Properties

Label 2-12e2-16.11-c2-0-17
Degree $2$
Conductor $144$
Sign $-0.944 + 0.328i$
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  + (1.13 − 1.64i)2-s + (−1.42 − 3.73i)4-s + (−2.41 − 2.41i)5-s − 11.8·7-s + (−7.77 − 1.90i)8-s + (−6.71 + 1.23i)10-s + (11.9 − 11.9i)11-s + (2.08 − 2.08i)13-s + (−13.4 + 19.5i)14-s + (−11.9 + 10.6i)16-s + 23.1·17-s + (−6.77 − 6.77i)19-s + (−5.58 + 12.4i)20-s + (−6.10 − 33.2i)22-s − 3.92·23-s + ⋯
L(s)  = 1  + (0.567 − 0.823i)2-s + (−0.355 − 0.934i)4-s + (−0.482 − 0.482i)5-s − 1.69·7-s + (−0.971 − 0.237i)8-s + (−0.671 + 0.123i)10-s + (1.08 − 1.08i)11-s + (0.160 − 0.160i)13-s + (−0.962 + 1.39i)14-s + (−0.746 + 0.664i)16-s + 1.36·17-s + (−0.356 − 0.356i)19-s + (−0.279 + 0.622i)20-s + (−0.277 − 1.51i)22-s − 0.170·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.944 + 0.328i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.944 + 0.328i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.204374 - 1.21008i\)
\(L(\frac12)\) \(\approx\) \(0.204374 - 1.21008i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 1.64i)T \)
3 \( 1 \)
good5 \( 1 + (2.41 + 2.41i)T + 25iT^{2} \)
7 \( 1 + 11.8T + 49T^{2} \)
11 \( 1 + (-11.9 + 11.9i)T - 121iT^{2} \)
13 \( 1 + (-2.08 + 2.08i)T - 169iT^{2} \)
17 \( 1 - 23.1T + 289T^{2} \)
19 \( 1 + (6.77 + 6.77i)T + 361iT^{2} \)
23 \( 1 + 3.92T + 529T^{2} \)
29 \( 1 + (0.782 - 0.782i)T - 841iT^{2} \)
31 \( 1 - 2.65iT - 961T^{2} \)
37 \( 1 + (-37.2 - 37.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 69.1iT - 1.68e3T^{2} \)
43 \( 1 + (29.2 - 29.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 68.0iT - 2.20e3T^{2} \)
53 \( 1 + (-40.3 - 40.3i)T + 2.80e3iT^{2} \)
59 \( 1 + (23.5 - 23.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (-65.9 + 65.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (40.9 + 40.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 98.1T + 5.04e3T^{2} \)
73 \( 1 - 74.2iT - 5.32e3T^{2} \)
79 \( 1 - 0.779iT - 6.24e3T^{2} \)
83 \( 1 + (-91.7 - 91.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 20.1iT - 7.92e3T^{2} \)
97 \( 1 + 86.6T + 9.40e3T^{2} \)
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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

−12.34338372737049984874719238391, −11.71437005464582865686311395811, −10.40371037573945230612069523388, −9.502955699087820482380651471004, −8.545184570141503483267705946938, −6.62158885446352732551126256517, −5.70109688428615435731383546880, −3.98002527869020576463111786634, −3.13284812126485089714724245348, −0.68768497778719670987154770425, 3.19847477562044003132214477516, 4.13274557279971261284940182371, 5.93609682151195581054688577535, 6.79262734503306866884169115151, 7.64229896460995130382328811072, 9.192701280239025980987037937317, 9.979331414237672962795665560799, 11.71277037746276527507599254606, 12.48666037440797412378681039424, 13.27306153111147290461127732355

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