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Properties

Label 2-12e2-144.131-c1-0-5
Degree $2$
Conductor $144$
Sign $0.561 - 0.827i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.311 + 1.37i)2-s + (−1.36 − 1.06i)3-s + (−1.80 − 0.858i)4-s + (2.31 − 0.619i)5-s + (1.89 − 1.55i)6-s + (2.51 + 4.35i)7-s + (1.74 − 2.22i)8-s + (0.733 + 2.90i)9-s + (0.135 + 3.38i)10-s + (−1.03 − 0.276i)11-s + (1.55 + 3.09i)12-s + (4.00 − 1.07i)13-s + (−6.78 + 2.11i)14-s + (−3.81 − 1.61i)15-s + (2.52 + 3.10i)16-s − 2.22i·17-s + ⋯
L(s)  = 1  + (−0.219 + 0.975i)2-s + (−0.788 − 0.614i)3-s + (−0.903 − 0.429i)4-s + (1.03 − 0.276i)5-s + (0.773 − 0.634i)6-s + (0.949 + 1.64i)7-s + (0.617 − 0.786i)8-s + (0.244 + 0.969i)9-s + (0.0427 + 1.06i)10-s + (−0.310 − 0.0833i)11-s + (0.448 + 0.893i)12-s + (1.11 − 0.297i)13-s + (−1.81 + 0.564i)14-s + (−0.985 − 0.416i)15-s + (0.631 + 0.775i)16-s − 0.539i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.814223 + 0.431385i\)
\(L(\frac12)\) \(\approx\) \(0.814223 + 0.431385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.311 - 1.37i)T \)
3 \( 1 + (1.36 + 1.06i)T \)
good5 \( 1 + (-2.31 + 0.619i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.51 - 4.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.03 + 0.276i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.00 + 1.07i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 2.22iT - 17T^{2} \)
19 \( 1 + (0.697 - 0.697i)T - 19iT^{2} \)
23 \( 1 + (2.20 + 1.27i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.589 - 0.157i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.190 - 0.109i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.16 - 5.16i)T - 37iT^{2} \)
41 \( 1 + (-0.828 + 1.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.33 + 4.98i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.76 + 9.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.80 + 7.80i)T + 53iT^{2} \)
59 \( 1 + (-1.36 - 5.09i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.73 + 6.48i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.20 + 8.22i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + (-7.74 + 4.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.48 - 5.55i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 6.56T + 89T^{2} \)
97 \( 1 + (1.51 + 2.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.37204637004516186639832527266, −12.42137181706127097685327111899, −11.29975776470268052940488686879, −10.04663877188632171474014025840, −8.767292289752980243200875209820, −8.094190371730043670012922181964, −6.50746821732284909206105843456, −5.62010443532444529864040713584, −5.10199268679097318727039382328, −1.80525785726359857899404836537, 1.44451580822680463203449719550, 3.77942470064949970809569654358, 4.78245994375795329945535886328, 6.21283283513354489766254723161, 7.80680844718232791816409396685, 9.250918391333685031694772746360, 10.34465791025165582490065607032, 10.71364398875723499479179700824, 11.50938584840679905058223200292, 12.93396322858860133676261848423

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