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Properties

Label 2-12e2-16.13-c1-0-0
Degree $2$
Conductor $144$
Sign $0.439 - 0.898i$
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  + (−0.767 − 1.18i)2-s + (−0.822 + 1.82i)4-s + (−2.37 + 2.37i)5-s + 3.64i·7-s + (2.79 − 0.420i)8-s + (4.64 + 0.999i)10-s + (−0.841 + 0.841i)11-s + (−2.64 − 2.64i)13-s + (4.33 − 2.79i)14-s + (−2.64 − 2.99i)16-s + 3.06·17-s + (1.64 + 1.64i)19-s + (−2.37 − 6.28i)20-s + (1.64 + 0.354i)22-s + 7.82i·23-s + ⋯
L(s)  = 1  + (−0.542 − 0.840i)2-s + (−0.411 + 0.911i)4-s + (−1.06 + 1.06i)5-s + 1.37i·7-s + (0.988 − 0.148i)8-s + (1.46 + 0.316i)10-s + (−0.253 + 0.253i)11-s + (−0.733 − 0.733i)13-s + (1.15 − 0.747i)14-s + (−0.661 − 0.749i)16-s + 0.744·17-s + (0.377 + 0.377i)19-s + (−0.531 − 1.40i)20-s + (0.350 + 0.0755i)22-s + 1.63i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.439 - 0.898i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.439 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489423 + 0.305289i\)
\(L(\frac12)\) \(\approx\) \(0.489423 + 0.305289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.767 + 1.18i)T \)
3 \( 1 \)
good5 \( 1 + (2.37 - 2.37i)T - 5iT^{2} \)
7 \( 1 - 3.64iT - 7T^{2} \)
11 \( 1 + (0.841 - 0.841i)T - 11iT^{2} \)
13 \( 1 + (2.64 + 2.64i)T + 13iT^{2} \)
17 \( 1 - 3.06T + 17T^{2} \)
19 \( 1 + (-1.64 - 1.64i)T + 19iT^{2} \)
23 \( 1 - 7.82iT - 23T^{2} \)
29 \( 1 + (0.692 + 0.692i)T + 29iT^{2} \)
31 \( 1 + 0.354T + 31T^{2} \)
37 \( 1 + (-4.64 + 4.64i)T - 37iT^{2} \)
41 \( 1 + 6.43iT - 41T^{2} \)
43 \( 1 + (5.64 - 5.64i)T - 43iT^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + (5.44 - 5.44i)T - 53iT^{2} \)
59 \( 1 + (-7.82 + 7.82i)T - 59iT^{2} \)
61 \( 1 + (-4.64 - 4.64i)T + 61iT^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 + 3.36iT - 71T^{2} \)
73 \( 1 - 7.29iT - 73T^{2} \)
79 \( 1 - 4.35T + 79T^{2} \)
83 \( 1 + (-0.841 - 0.841i)T + 83iT^{2} \)
89 \( 1 + 9.50iT - 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−12.81519280780010891825553237631, −11.98571252310222009920302778986, −11.40936132975467720203871848397, −10.29010152048716074543334840084, −9.330828081207855854985117384678, −7.986131268028729509577513602948, −7.37005059305740434651828709466, −5.44902814528344634743732946284, −3.60227794843516931200467080590, −2.56280654812003011364367156303, 0.71150562094997107158395850868, 4.15376254940790847285444778108, 4.99308575855915356715842320538, 6.78732685017159871378205975408, 7.69612803461793922242960353307, 8.477918369327214844035293682933, 9.666616935468842334274263721944, 10.70119640640942019232898899398, 11.89519017144440000201295813339, 13.07666969320014885325588445676

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