Properties

Label 2-12e2-144.85-c1-0-4
Degree $2$
Conductor $144$
Sign $0.445 - 0.895i$
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  + (−1.18 + 0.777i)2-s + (−0.409 + 1.68i)3-s + (0.791 − 1.83i)4-s + (3.30 − 0.884i)5-s + (−0.824 − 2.30i)6-s + (2.63 − 1.51i)7-s + (0.492 + 2.78i)8-s + (−2.66 − 1.37i)9-s + (−3.21 + 3.61i)10-s + (−1.39 + 5.21i)11-s + (2.76 + 2.08i)12-s + (−0.378 − 1.41i)13-s + (−1.92 + 3.84i)14-s + (0.137 + 5.91i)15-s + (−2.74 − 2.90i)16-s + 0.259·17-s + ⋯
L(s)  = 1  + (−0.835 + 0.549i)2-s + (−0.236 + 0.971i)3-s + (0.395 − 0.918i)4-s + (1.47 − 0.395i)5-s + (−0.336 − 0.941i)6-s + (0.994 − 0.574i)7-s + (0.174 + 0.984i)8-s + (−0.888 − 0.459i)9-s + (−1.01 + 1.14i)10-s + (−0.421 + 1.57i)11-s + (0.798 + 0.601i)12-s + (−0.104 − 0.391i)13-s + (−0.515 + 1.02i)14-s + (0.0355 + 1.52i)15-s + (−0.686 − 0.726i)16-s + 0.0629·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.445 - 0.895i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.445 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772207 + 0.478240i\)
\(L(\frac12)\) \(\approx\) \(0.772207 + 0.478240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 - 0.777i)T \)
3 \( 1 + (0.409 - 1.68i)T \)
good5 \( 1 + (-3.30 + 0.884i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.63 + 1.51i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.39 - 5.21i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.378 + 1.41i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 0.259T + 17T^{2} \)
19 \( 1 + (-0.228 + 0.228i)T - 19iT^{2} \)
23 \( 1 + (2.69 + 1.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.63 - 0.438i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.30 - 5.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.24 - 1.24i)T + 37iT^{2} \)
41 \( 1 + (8.85 + 5.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.722 + 2.69i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.24 - 1.24i)T + 53iT^{2} \)
59 \( 1 + (-0.725 + 0.194i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.36 - 1.16i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.411 - 1.53i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.68iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 + (0.738 + 1.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.39 + 0.908i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 + (-5.94 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.62092157459901608540690678513, −12.07075081924041464856952340761, −10.57345104848180435610206179467, −10.19299577351207413038038235831, −9.333387778725946635522175046243, −8.244122291807524519495885087858, −6.85595985037607612539259448997, −5.42197415493399496877820237655, −4.80775769761709192908332496651, −1.92057467227424376178253690742, 1.63654425188774709619948427296, 2.74047911727774459899836995112, 5.54879741350637546412192594946, 6.44407736794078504190761832583, 7.893083250140172347731069248431, 8.689202464628616782307780381623, 9.875708747303999214858101095333, 11.11015390517426216069669941555, 11.56495063666438388105225557777, 12.92100116938104468940919204668

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