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Properties

Label 2-12e2-144.59-c1-0-14
Degree $2$
Conductor $144$
Sign $0.529 + 0.848i$
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  + (−1.41 − 0.0478i)2-s + (1.72 + 0.117i)3-s + (1.99 + 0.135i)4-s + (−0.961 − 3.58i)5-s + (−2.43 − 0.249i)6-s + (−1.29 − 2.23i)7-s + (−2.81 − 0.286i)8-s + (2.97 + 0.406i)9-s + (1.18 + 5.11i)10-s + (−0.541 + 2.02i)11-s + (3.43 + 0.468i)12-s + (0.267 + 0.998i)13-s + (1.71 + 3.22i)14-s + (−1.23 − 6.31i)15-s + (3.96 + 0.539i)16-s − 4.13i·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0338i)2-s + (0.997 + 0.0679i)3-s + (0.997 + 0.0676i)4-s + (−0.430 − 1.60i)5-s + (−0.994 − 0.101i)6-s + (−0.488 − 0.845i)7-s + (−0.994 − 0.101i)8-s + (0.990 + 0.135i)9-s + (0.375 + 1.61i)10-s + (−0.163 + 0.609i)11-s + (0.990 + 0.135i)12-s + (0.0741 + 0.276i)13-s + (0.459 + 0.861i)14-s + (−0.319 − 1.63i)15-s + (0.990 + 0.134i)16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.529 + 0.848i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.529 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.793032 - 0.440048i\)
\(L(\frac12)\) \(\approx\) \(0.793032 - 0.440048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0478i)T \)
3 \( 1 + (-1.72 - 0.117i)T \)
good5 \( 1 + (0.961 + 3.58i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.29 + 2.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.541 - 2.02i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.267 - 0.998i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 4.13iT - 17T^{2} \)
19 \( 1 + (-2.49 - 2.49i)T + 19iT^{2} \)
23 \( 1 + (-7.01 - 4.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.88 - 7.02i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.03 - 1.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.75 + 4.75i)T + 37iT^{2} \)
41 \( 1 + (0.636 - 1.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.45 + 0.389i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.60 - 6.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.546 - 0.546i)T - 53iT^{2} \)
59 \( 1 + (8.00 - 2.14i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.77 + 1.81i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.80 - 0.751i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.59iT - 71T^{2} \)
73 \( 1 + 8.78iT - 73T^{2} \)
79 \( 1 + (5.64 - 3.26i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.71 - 2.06i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + (-6.54 - 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.83926859051600618291975654157, −12.04099345594435570965109417646, −10.58370753362977988898950714888, −9.347840891207066937991092708286, −9.070963107338300555353313654634, −7.76034778423418675524200676191, −7.13874914128611921399311098413, −4.93008377596072301303401090200, −3.42087241953978100349510030742, −1.30541603910930620722187590742, 2.56969134949188532679411632031, 3.31369962115810353041622269768, 6.18645043124642951082551218424, 7.08741303987390752345615687542, 8.111350463814867041773319242904, 9.022423791715510404961787510681, 10.15186574016813148993804848899, 10.92408585360724659651142237698, 12.02826267590967071565685051238, 13.34412951524869445427259869931

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