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Properties

Label 2-12e2-144.133-c1-0-18
Degree $2$
Conductor $144$
Sign $0.461 + 0.887i$
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  + (−0.366 + 1.36i)2-s − 1.73i·3-s + (−1.73 − i)4-s + (0.5 − 1.86i)5-s + (2.36 + 0.633i)6-s + (−3.86 − 2.23i)7-s + (2 − 1.99i)8-s − 2.99·9-s + (2.36 + 1.36i)10-s + (1.86 − 0.5i)11-s + (−1.73 + 2.99i)12-s + (2.23 + 0.598i)13-s + (4.46 − 4.46i)14-s + (−3.23 − 0.866i)15-s + (1.99 + 3.46i)16-s + 4·17-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s − 0.999i·3-s + (−0.866 − 0.5i)4-s + (0.223 − 0.834i)5-s + (0.965 + 0.258i)6-s + (−1.46 − 0.843i)7-s + (0.707 − 0.707i)8-s − 0.999·9-s + (0.748 + 0.431i)10-s + (0.562 − 0.150i)11-s + (−0.499 + 0.866i)12-s + (0.619 + 0.165i)13-s + (1.19 − 1.19i)14-s + (−0.834 − 0.223i)15-s + (0.499 + 0.866i)16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.461 + 0.887i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.461 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663983 - 0.402914i\)
\(L(\frac12)\) \(\approx\) \(0.663983 - 0.402914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + 1.73iT \)
good5 \( 1 + (-0.5 + 1.86i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (3.86 + 2.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.86 + 0.5i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-2.23 - 0.598i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + (-5.59 + 3.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.232 - 0.866i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.59 + 7.96i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.26 + 4.26i)T + 37iT^{2} \)
41 \( 1 + (0.696 - 0.401i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.33 + 1.69i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.598 - 1.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.73 - 5.73i)T + 53iT^{2} \)
59 \( 1 + (-0.401 + 1.5i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.571 - 2.13i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-8.33 - 2.23i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 + 7.46iT - 73T^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.79 - 14.1i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 15.8iT - 89T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.99867465693642473685032678544, −12.51674886113895300814304979273, −10.70885153941083645267889106240, −9.431577443043981254048006040540, −8.674970061269965289316890143279, −7.42532181144775920698853247253, −6.52813017654069927913816182951, −5.65611816523331335781761941739, −3.80806570891253224534312301153, −0.913121273494331531416946209800, 2.86516634859788755610611831208, 3.58274782711945581778145147432, 5.36116064237471455353744662941, 6.70775926826863527110384410604, 8.699291742906490160131586632283, 9.377429014192716184212136941679, 10.24018892755385668166855631815, 11.01899735195603900907673627857, 12.10597555208484007716368641119, 13.07732750753171736091682875281

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