Properties

Label 2-12e2-3.2-c6-0-5
Degree $2$
Conductor $144$
Sign $0.577 - 0.816i$
Analytic cond. $33.1277$
Root an. cond. $5.75567$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 173. i·5-s + 484·7-s − 1.34e3i·11-s + 3.36e3·13-s + 12.7i·17-s − 5.74e3·19-s + 3.37e3i·23-s − 1.46e4·25-s + 2.93e4i·29-s + 3.97e4·31-s + 8.41e4i·35-s + 5.25e4·37-s − 3.70e4i·41-s − 3.80e3·43-s − 7.67e4i·47-s + ⋯
L(s)  = 1  + 1.39i·5-s + 1.41·7-s − 1.00i·11-s + 1.53·13-s + 0.00259i·17-s − 0.837·19-s + 0.277i·23-s − 0.936·25-s + 1.20i·29-s + 1.33·31-s + 1.96i·35-s + 1.03·37-s − 0.537i·41-s − 0.0477·43-s − 0.739i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(33.1277\)
Root analytic conductor: \(5.75567\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.510495069\)
\(L(\frac12)\) \(\approx\) \(2.510495069\)
\(L(4)\) 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 \( 1 \)
good5 \( 1 - 173. iT - 1.56e4T^{2} \)
7 \( 1 - 484T + 1.17e5T^{2} \)
11 \( 1 + 1.34e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.36e3T + 4.82e6T^{2} \)
17 \( 1 - 12.7iT - 2.41e7T^{2} \)
19 \( 1 + 5.74e3T + 4.70e7T^{2} \)
23 \( 1 - 3.37e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.93e4iT - 5.94e8T^{2} \)
31 \( 1 - 3.97e4T + 8.87e8T^{2} \)
37 \( 1 - 5.25e4T + 2.56e9T^{2} \)
41 \( 1 + 3.70e4iT - 4.75e9T^{2} \)
43 \( 1 + 3.80e3T + 6.32e9T^{2} \)
47 \( 1 + 7.67e4iT - 1.07e10T^{2} \)
53 \( 1 - 2.38e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.49e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.32e4T + 5.15e10T^{2} \)
67 \( 1 + 1.68e5T + 9.04e10T^{2} \)
71 \( 1 - 5.31e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.36e5T + 1.51e11T^{2} \)
79 \( 1 - 3.51e4T + 2.43e11T^{2} \)
83 \( 1 - 1.09e4iT - 3.26e11T^{2} \)
89 \( 1 + 1.29e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.21e5T + 8.32e11T^{2} \)
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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

−11.75658301839717524534462347246, −10.94325142230083535338597538769, −10.58971707827712798579318381850, −8.772853546570716716963485991316, −7.982442961510968947010264665721, −6.70100945721013363536436576813, −5.70754577173580074259172128053, −4.07536074456574609125939238883, −2.80971070822736746315287972799, −1.27129565460516531850455623138, 0.916004814440346501894964494509, 1.92145876805885855154170684455, 4.25270911390618045706173370341, 4.87842778730165009840672362270, 6.24352548407250608505165201742, 8.015378540875718728436061159478, 8.443101918666047310890345475419, 9.621784142988911707236896353435, 10.96299692797315305569050367123, 11.83797341295481120918366888217

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