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Properties

Label 2-12e2-48.11-c3-0-10
Degree $2$
Conductor $144$
Sign $0.148 - 0.988i$
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  + (1.80 + 2.17i)2-s + (−1.47 + 7.86i)4-s + (6.31 − 6.31i)5-s + 16.2·7-s + (−19.7 + 10.9i)8-s + (25.1 + 2.34i)10-s + (48.1 + 48.1i)11-s + (−8.61 + 8.61i)13-s + (29.3 + 35.4i)14-s + (−59.6 − 23.2i)16-s + 53.2i·17-s + (−55.5 − 55.5i)19-s + (40.3 + 58.9i)20-s + (−17.8 + 191. i)22-s − 66.9i·23-s + ⋯
L(s)  = 1  + (0.638 + 0.769i)2-s + (−0.184 + 0.982i)4-s + (0.564 − 0.564i)5-s + 0.878·7-s + (−0.874 + 0.485i)8-s + (0.795 + 0.0741i)10-s + (1.31 + 1.31i)11-s + (−0.183 + 0.183i)13-s + (0.561 + 0.676i)14-s + (−0.931 − 0.363i)16-s + 0.759i·17-s + (−0.670 − 0.670i)19-s + (0.450 + 0.659i)20-s + (−0.173 + 1.85i)22-s − 0.607i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ 0.148 - 0.988i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.01162 + 1.73239i\)
\(L(\frac12)\) \(\approx\) \(2.01162 + 1.73239i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.80 - 2.17i)T \)
3 \( 1 \)
good5 \( 1 + (-6.31 + 6.31i)T - 125iT^{2} \)
7 \( 1 - 16.2T + 343T^{2} \)
11 \( 1 + (-48.1 - 48.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (8.61 - 8.61i)T - 2.19e3iT^{2} \)
17 \( 1 - 53.2iT - 4.91e3T^{2} \)
19 \( 1 + (55.5 + 55.5i)T + 6.85e3iT^{2} \)
23 \( 1 + 66.9iT - 1.21e4T^{2} \)
29 \( 1 + (-126. - 126. i)T + 2.43e4iT^{2} \)
31 \( 1 + 121. iT - 2.97e4T^{2} \)
37 \( 1 + (250. + 250. i)T + 5.06e4iT^{2} \)
41 \( 1 - 402.T + 6.89e4T^{2} \)
43 \( 1 + (-187. + 187. i)T - 7.95e4iT^{2} \)
47 \( 1 + 96.1T + 1.03e5T^{2} \)
53 \( 1 + (90.3 - 90.3i)T - 1.48e5iT^{2} \)
59 \( 1 + (488. + 488. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-378. + 378. i)T - 2.26e5iT^{2} \)
67 \( 1 + (223. + 223. i)T + 3.00e5iT^{2} \)
71 \( 1 + 231. iT - 3.57e5T^{2} \)
73 \( 1 - 265. iT - 3.89e5T^{2} \)
79 \( 1 + 604. iT - 4.93e5T^{2} \)
83 \( 1 + (-351. + 351. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + 1.85e3T + 9.12e5T^{2} \)
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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

−12.78233142922539072603728343936, −12.29185158994305628862854397591, −11.03524603451098944267705698073, −9.409604714479263667582359540061, −8.637446218512164864518907736253, −7.34203595416474580862327340229, −6.31444289009987180663658144137, −4.97106596695369786033997674594, −4.16072037204956643296889651405, −1.91111814540366918457027653070, 1.29235609297341535935916922884, 2.86229109119369956048318180331, 4.26126036108701844450709213678, 5.67892928857725282999058807080, 6.59838282311370264762063062856, 8.401885852218813161315299742846, 9.554979171854107716691177336610, 10.63105918624607193205097876534, 11.43235911757680618224442713768, 12.19931794873470376403065394480

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