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Properties

Label 2-12e2-36.31-c2-0-6
Degree $2$
Conductor $144$
Sign $0.761 + 0.648i$
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  + (−2.82 + 1.01i)3-s + (1.81 − 3.13i)5-s + (−1.59 + 0.920i)7-s + (6.93 − 5.73i)9-s + (10.0 − 5.80i)11-s + (6.43 − 11.1i)13-s + (−1.92 + 10.6i)15-s + 12.6·17-s − 25.6i·19-s + (3.56 − 4.21i)21-s + (25.9 + 14.9i)23-s + (5.93 + 10.2i)25-s + (−13.7 + 23.2i)27-s + (−10.8 − 18.7i)29-s + (−52.4 − 30.2i)31-s + ⋯
L(s)  = 1  + (−0.940 + 0.338i)3-s + (0.362 − 0.627i)5-s + (−0.227 + 0.131i)7-s + (0.770 − 0.637i)9-s + (0.914 − 0.528i)11-s + (0.495 − 0.857i)13-s + (−0.128 + 0.713i)15-s + 0.742·17-s − 1.35i·19-s + (0.169 − 0.200i)21-s + (1.12 + 0.650i)23-s + (0.237 + 0.411i)25-s + (−0.509 + 0.860i)27-s + (−0.372 − 0.645i)29-s + (−1.69 − 0.975i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.761 + 0.648i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.761 + 0.648i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07220 - 0.394556i\)
\(L(\frac12)\) \(\approx\) \(1.07220 - 0.394556i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.82 - 1.01i)T \)
good5 \( 1 + (-1.81 + 3.13i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (1.59 - 0.920i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-10.0 + 5.80i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-6.43 + 11.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 12.6T + 289T^{2} \)
19 \( 1 + 25.6iT - 361T^{2} \)
23 \( 1 + (-25.9 - 14.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (10.8 + 18.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (52.4 + 30.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 25.7T + 1.36e3T^{2} \)
41 \( 1 + (33.3 - 57.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-14.2 + 8.22i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-66.1 + 38.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 14.2T + 2.80e3T^{2} \)
59 \( 1 + (-50.3 - 29.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (9.43 + 16.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-20.6 - 11.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 46.4iT - 5.04e3T^{2} \)
73 \( 1 - 49.3T + 5.32e3T^{2} \)
79 \( 1 + (52.4 - 30.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (86.2 - 49.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 154.T + 7.92e3T^{2} \)
97 \( 1 + (21.1 + 36.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.78766031408470831583159386521, −11.60603894744029970576062542846, −10.89715394127719572289786198609, −9.597659792066135445444131355083, −8.894457828536137550426953702590, −7.20536064847101307819329990240, −5.92645790724763940563225747599, −5.13712866253830815454553799786, −3.58421512425179137672858490194, −0.968507342595558254918063923953, 1.62669349992560679914981584038, 3.81957729028955871820072175357, 5.41430279139394514522287634823, 6.58516944110582325904884349072, 7.20369602683073081771368411946, 8.934490441772733711123427213319, 10.18454736242680272387209434603, 10.90507240589649111908761629950, 12.05310510002668846160255500400, 12.70719122110043174674438803346

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