Properties

Label 2-12e2-36.31-c2-0-10
Degree $2$
Conductor $144$
Sign $0.761 + 0.648i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.82 − 1.01i)3-s + (1.81 − 3.13i)5-s + (1.59 − 0.920i)7-s + (6.93 − 5.73i)9-s + (−10.0 + 5.80i)11-s + (6.43 − 11.1i)13-s + (1.92 − 10.6i)15-s + 12.6·17-s + 25.6i·19-s + (3.56 − 4.21i)21-s + (−25.9 − 14.9i)23-s + (5.93 + 10.2i)25-s + (13.7 − 23.2i)27-s + (−10.8 − 18.7i)29-s + (52.4 + 30.2i)31-s + ⋯
L(s)  = 1  + (0.940 − 0.338i)3-s + (0.362 − 0.627i)5-s + (0.227 − 0.131i)7-s + (0.770 − 0.637i)9-s + (−0.914 + 0.528i)11-s + (0.495 − 0.857i)13-s + (0.128 − 0.713i)15-s + 0.742·17-s + 1.35i·19-s + (0.169 − 0.200i)21-s + (−1.12 − 0.650i)23-s + (0.237 + 0.411i)25-s + (0.509 − 0.860i)27-s + (−0.372 − 0.645i)29-s + (1.69 + 0.975i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.761 + 0.648i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.761 + 0.648i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.88708 - 0.694423i\)
\(L(\frac12)\) \(\approx\) \(1.88708 - 0.694423i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.82 + 1.01i)T \)
good5 \( 1 + (-1.81 + 3.13i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-1.59 + 0.920i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (10.0 - 5.80i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-6.43 + 11.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 12.6T + 289T^{2} \)
19 \( 1 - 25.6iT - 361T^{2} \)
23 \( 1 + (25.9 + 14.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (10.8 + 18.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-52.4 - 30.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 25.7T + 1.36e3T^{2} \)
41 \( 1 + (33.3 - 57.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.2 - 8.22i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (66.1 - 38.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 14.2T + 2.80e3T^{2} \)
59 \( 1 + (50.3 + 29.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (9.43 + 16.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (20.6 + 11.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 46.4iT - 5.04e3T^{2} \)
73 \( 1 - 49.3T + 5.32e3T^{2} \)
79 \( 1 + (-52.4 + 30.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-86.2 + 49.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 154.T + 7.92e3T^{2} \)
97 \( 1 + (21.1 + 36.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.83622491407295972505578669935, −12.13467249905801655572790378344, −10.36098692112488974444402288595, −9.716994910144185414306164266046, −8.214426904851033359660940697983, −7.917967919738943435559823689127, −6.22478481116022580273302550809, −4.81167542434205107467223200960, −3.20583851417540534957264190261, −1.53834626689562160212223333482, 2.23206819930170172164549695027, 3.50471574577258872599269973448, 5.06362341924279876738770708971, 6.59778619259477448618819794035, 7.85436404984490598872586823271, 8.793922843186528160585287609200, 9.918846359101041184416915633274, 10.72843960494289586432064790915, 11.89477452604443543399310303037, 13.54022299250575324439033397104

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