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Properties

Label 2-12e2-48.35-c3-0-15
Degree $2$
Conductor $144$
Sign $0.995 - 0.0959i$
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  + (2.79 − 0.463i)2-s + (7.56 − 2.58i)4-s + (9.40 + 9.40i)5-s − 3.57·7-s + (19.9 − 10.7i)8-s + (30.5 + 21.8i)10-s + (3.36 − 3.36i)11-s + (26.9 + 26.9i)13-s + (−9.96 + 1.65i)14-s + (50.6 − 39.1i)16-s + 12.7i·17-s + (−50.0 + 50.0i)19-s + (95.4 + 46.8i)20-s + (7.82 − 10.9i)22-s − 208. i·23-s + ⋯
L(s)  = 1  + (0.986 − 0.163i)2-s + (0.946 − 0.323i)4-s + (0.840 + 0.840i)5-s − 0.192·7-s + (0.880 − 0.474i)8-s + (0.967 + 0.691i)10-s + (0.0921 − 0.0921i)11-s + (0.574 + 0.574i)13-s + (−0.190 + 0.0316i)14-s + (0.790 − 0.612i)16-s + 0.182i·17-s + (−0.604 + 0.604i)19-s + (1.06 + 0.523i)20-s + (0.0757 − 0.106i)22-s − 1.88i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0959i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.995 - 0.0959i$
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: \((2,\ 144,\ (\ :3/2),\ 0.995 - 0.0959i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.40973 + 0.164023i\)
\(L(\frac12)\) \(\approx\) \(3.40973 + 0.164023i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.79 + 0.463i)T \)
3 \( 1 \)
good5 \( 1 + (-9.40 - 9.40i)T + 125iT^{2} \)
7 \( 1 + 3.57T + 343T^{2} \)
11 \( 1 + (-3.36 + 3.36i)T - 1.33e3iT^{2} \)
13 \( 1 + (-26.9 - 26.9i)T + 2.19e3iT^{2} \)
17 \( 1 - 12.7iT - 4.91e3T^{2} \)
19 \( 1 + (50.0 - 50.0i)T - 6.85e3iT^{2} \)
23 \( 1 + 208. iT - 1.21e4T^{2} \)
29 \( 1 + (-134. + 134. i)T - 2.43e4iT^{2} \)
31 \( 1 - 80.1iT - 2.97e4T^{2} \)
37 \( 1 + (308. - 308. i)T - 5.06e4iT^{2} \)
41 \( 1 + 172.T + 6.89e4T^{2} \)
43 \( 1 + (87.0 + 87.0i)T + 7.95e4iT^{2} \)
47 \( 1 + 525.T + 1.03e5T^{2} \)
53 \( 1 + (-127. - 127. i)T + 1.48e5iT^{2} \)
59 \( 1 + (172. - 172. i)T - 2.05e5iT^{2} \)
61 \( 1 + (332. + 332. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-556. + 556. i)T - 3.00e5iT^{2} \)
71 \( 1 + 450. iT - 3.57e5T^{2} \)
73 \( 1 - 797. iT - 3.89e5T^{2} \)
79 \( 1 + 70.1iT - 4.93e5T^{2} \)
83 \( 1 + (636. + 636. i)T + 5.71e5iT^{2} \)
89 \( 1 - 925.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−12.76130226706502795153878611521, −11.75756026948472055804953624706, −10.59756377534580993331947470478, −10.05388369754517297872478056931, −8.380475380752301890008934789407, −6.55258491011054750344227311842, −6.34896924174736241406957290554, −4.71305613764266066116488429080, −3.25301743748347847860083509335, −1.93618251710940251915651202009, 1.65289231889466753352910011641, 3.39156907790067387678674355872, 4.94193008919637789180148313974, 5.76076031357482614335340075737, 6.95385425118522634204047017399, 8.351055613353281764592645080839, 9.534841351877770665014200098683, 10.76354570912106479172019116821, 11.86431234797947421546815777028, 13.00306407740773105602185735903

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