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Properties

Label 2-12e2-48.35-c3-0-0
Degree $2$
Conductor $144$
Sign $-0.714 - 0.700i$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.89 − 2.10i)2-s + (−0.849 + 7.95i)4-s + (11.2 + 11.2i)5-s − 19.1·7-s + (18.3 − 13.2i)8-s + (2.40 − 45.1i)10-s + (−34.3 + 34.3i)11-s + (−59.4 − 59.4i)13-s + (36.3 + 40.3i)14-s + (−62.5 − 13.5i)16-s − 102. i·17-s + (−60.4 + 60.4i)19-s + (−99.4 + 80.2i)20-s + (137. + 7.30i)22-s + 106. i·23-s + ⋯
L(s)  = 1  + (−0.668 − 0.743i)2-s + (−0.106 + 0.994i)4-s + (1.01 + 1.01i)5-s − 1.03·7-s + (0.810 − 0.585i)8-s + (0.0759 − 1.42i)10-s + (−0.941 + 0.941i)11-s + (−1.26 − 1.26i)13-s + (0.692 + 0.770i)14-s + (−0.977 − 0.211i)16-s − 1.46i·17-s + (−0.729 + 0.729i)19-s + (−1.11 + 0.897i)20-s + (1.32 + 0.0707i)22-s + 0.961i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.700i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.714 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.714 - 0.700i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ -0.714 - 0.700i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.112408 + 0.275243i\)
\(L(\frac12)\) \(\approx\) \(0.112408 + 0.275243i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.89 + 2.10i)T \)
3 \( 1 \)
good5 \( 1 + (-11.2 - 11.2i)T + 125iT^{2} \)
7 \( 1 + 19.1T + 343T^{2} \)
11 \( 1 + (34.3 - 34.3i)T - 1.33e3iT^{2} \)
13 \( 1 + (59.4 + 59.4i)T + 2.19e3iT^{2} \)
17 \( 1 + 102. iT - 4.91e3T^{2} \)
19 \( 1 + (60.4 - 60.4i)T - 6.85e3iT^{2} \)
23 \( 1 - 106. iT - 1.21e4T^{2} \)
29 \( 1 + (30.6 - 30.6i)T - 2.43e4iT^{2} \)
31 \( 1 - 99.7iT - 2.97e4T^{2} \)
37 \( 1 + (94.9 - 94.9i)T - 5.06e4iT^{2} \)
41 \( 1 + 34.5T + 6.89e4T^{2} \)
43 \( 1 + (198. + 198. i)T + 7.95e4iT^{2} \)
47 \( 1 - 314.T + 1.03e5T^{2} \)
53 \( 1 + (187. + 187. i)T + 1.48e5iT^{2} \)
59 \( 1 + (382. - 382. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-509. - 509. i)T + 2.26e5iT^{2} \)
67 \( 1 + (107. - 107. i)T - 3.00e5iT^{2} \)
71 \( 1 + 209. iT - 3.57e5T^{2} \)
73 \( 1 - 411. iT - 3.89e5T^{2} \)
79 \( 1 + 992. iT - 4.93e5T^{2} \)
83 \( 1 + (-852. - 852. i)T + 5.71e5iT^{2} \)
89 \( 1 - 205.T + 7.04e5T^{2} \)
97 \( 1 + 775.T + 9.12e5T^{2} \)
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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

−12.88620152681528712690959888479, −12.03035421395772486243679448803, −10.42742283958869625253711691663, −10.14623973518999597453225431191, −9.377665113223135331050598574441, −7.64450378423982075747694826953, −6.87279445007109993547156713713, −5.25780003040773158147689853460, −3.12260191467828013828113951560, −2.33328357152901570625129801853, 0.16449755422850139482577074010, 2.11976315559880473534055537088, 4.67493598303565913633386520488, 5.86044902712137694833684576278, 6.67597349990051170520251038151, 8.231357866569816141878807865023, 9.130967054863083674211111689426, 9.839659931739331459300260824684, 10.83570925103388295487419344551, 12.59374656284749501095207842702

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