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Properties

Label 2-12e2-16.3-c4-0-16
Degree $2$
Conductor $144$
Sign $0.607 - 0.794i$
Analytic cond. $14.8852$
Root an. cond. $3.85814$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.96 − 0.523i)2-s + (15.4 − 4.15i)4-s + (−21.7 + 21.7i)5-s − 6.62·7-s + (59.1 − 24.5i)8-s + (−74.8 + 97.5i)10-s + (90.9 + 90.9i)11-s + (221. + 221. i)13-s + (−26.2 + 3.46i)14-s + (221. − 128. i)16-s + 132.·17-s + (−402. + 402. i)19-s + (−245. + 426. i)20-s + (408. + 313. i)22-s − 27.5·23-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (0.965 − 0.259i)4-s + (−0.869 + 0.869i)5-s − 0.135·7-s + (0.923 − 0.383i)8-s + (−0.748 + 0.975i)10-s + (0.752 + 0.752i)11-s + (1.31 + 1.31i)13-s + (−0.133 + 0.0176i)14-s + (0.865 − 0.501i)16-s + 0.458·17-s + (−1.11 + 1.11i)19-s + (−0.614 + 1.06i)20-s + (0.843 + 0.647i)22-s − 0.0519·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(14.8852\)
Root analytic conductor: \(3.85814\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :2),\ 0.607 - 0.794i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.72326 + 1.34522i\)
\(L(\frac12)\) \(\approx\) \(2.72326 + 1.34522i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.96 + 0.523i)T \)
3 \( 1 \)
good5 \( 1 + (21.7 - 21.7i)T - 625iT^{2} \)
7 \( 1 + 6.62T + 2.40e3T^{2} \)
11 \( 1 + (-90.9 - 90.9i)T + 1.46e4iT^{2} \)
13 \( 1 + (-221. - 221. i)T + 2.85e4iT^{2} \)
17 \( 1 - 132.T + 8.35e4T^{2} \)
19 \( 1 + (402. - 402. i)T - 1.30e5iT^{2} \)
23 \( 1 + 27.5T + 2.79e5T^{2} \)
29 \( 1 + (174. + 174. i)T + 7.07e5iT^{2} \)
31 \( 1 + 1.08e3iT - 9.23e5T^{2} \)
37 \( 1 + (-553. + 553. i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.80e3iT - 2.82e6T^{2} \)
43 \( 1 + (-17.8 - 17.8i)T + 3.41e6iT^{2} \)
47 \( 1 + 2.26e3iT - 4.87e6T^{2} \)
53 \( 1 + (-822. + 822. i)T - 7.89e6iT^{2} \)
59 \( 1 + (-972. - 972. i)T + 1.21e7iT^{2} \)
61 \( 1 + (2.05e3 + 2.05e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (-4.61e3 + 4.61e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 3.10e3T + 2.54e7T^{2} \)
73 \( 1 - 723. iT - 2.83e7T^{2} \)
79 \( 1 + 3.41e3iT - 3.89e7T^{2} \)
83 \( 1 + (-161. + 161. i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.46e3iT - 6.27e7T^{2} \)
97 \( 1 + 8.26e3T + 8.85e7T^{2} \)
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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

−12.45993206373191003700180259423, −11.57151000329433025207089273208, −10.96936225866339870327517282532, −9.711199633165502050968234263823, −8.050152807838408893032852611948, −6.85771411651465802303661138694, −6.15040186016469086740008642125, −4.23727420041074853834776132145, −3.61255961402248080474466880178, −1.83984781577586061316061931315, 0.961242237147534793497763818875, 3.22431547566672980856933053650, 4.22276298879891225570985288240, 5.49266890677939972061185450602, 6.61006003321596133341916589377, 8.051309777726710909364692499483, 8.756460724042402159904177126091, 10.67178662354777755628355559561, 11.42498813540894566219171801053, 12.49641976373538215893955184419

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