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Properties

Label 2-12e2-16.11-c2-0-7
Degree $2$
Conductor $144$
Sign $0.829 + 0.558i$
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  + (0.0999 − 1.99i)2-s + (−3.98 − 0.399i)4-s + (6.01 + 6.01i)5-s + 8.23·7-s + (−1.19 + 7.91i)8-s + (12.6 − 11.4i)10-s + (−6.51 + 6.51i)11-s + (8.82 − 8.82i)13-s + (0.822 − 16.4i)14-s + (15.6 + 3.17i)16-s + 14.1·17-s + (−23.7 − 23.7i)19-s + (−21.5 − 26.3i)20-s + (12.3 + 13.6i)22-s + 9.42·23-s + ⋯
L(s)  = 1  + (0.0499 − 0.998i)2-s + (−0.995 − 0.0997i)4-s + (1.20 + 1.20i)5-s + 1.17·7-s + (−0.149 + 0.988i)8-s + (1.26 − 1.14i)10-s + (−0.592 + 0.592i)11-s + (0.679 − 0.679i)13-s + (0.0587 − 1.17i)14-s + (0.980 + 0.198i)16-s + 0.833·17-s + (−1.24 − 1.24i)19-s + (−1.07 − 1.31i)20-s + (0.561 + 0.621i)22-s + 0.409·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.829 + 0.558i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.829 + 0.558i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64326 - 0.501693i\)
\(L(\frac12)\) \(\approx\) \(1.64326 - 0.501693i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0999 + 1.99i)T \)
3 \( 1 \)
good5 \( 1 + (-6.01 - 6.01i)T + 25iT^{2} \)
7 \( 1 - 8.23T + 49T^{2} \)
11 \( 1 + (6.51 - 6.51i)T - 121iT^{2} \)
13 \( 1 + (-8.82 + 8.82i)T - 169iT^{2} \)
17 \( 1 - 14.1T + 289T^{2} \)
19 \( 1 + (23.7 + 23.7i)T + 361iT^{2} \)
23 \( 1 - 9.42T + 529T^{2} \)
29 \( 1 + (23.7 - 23.7i)T - 841iT^{2} \)
31 \( 1 + 24.4iT - 961T^{2} \)
37 \( 1 + (-24.2 - 24.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 6.67iT - 1.68e3T^{2} \)
43 \( 1 + (-0.897 + 0.897i)T - 1.84e3iT^{2} \)
47 \( 1 + 25.2iT - 2.20e3T^{2} \)
53 \( 1 + (32.6 + 32.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (-8.31 + 8.31i)T - 3.48e3iT^{2} \)
61 \( 1 + (68.4 - 68.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (7.71 + 7.71i)T + 4.48e3iT^{2} \)
71 \( 1 + 137.T + 5.04e3T^{2} \)
73 \( 1 + 52.8iT - 5.32e3T^{2} \)
79 \( 1 + 87.2iT - 6.24e3T^{2} \)
83 \( 1 + (9.53 + 9.53i)T + 6.88e3iT^{2} \)
89 \( 1 + 146. iT - 7.92e3T^{2} \)
97 \( 1 - 101.T + 9.40e3T^{2} \)
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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

−12.95163771873903954655333314192, −11.41678742480904718109292308849, −10.73071983001789624968614243239, −10.09399434880089963574639058685, −8.865608724129739257177053586796, −7.56228653532288154920259383767, −5.94908756537286483785745137640, −4.81207769976045376223326235783, −2.97371615352683493673251055572, −1.81587443209288752247900003419, 1.47244551226660127964620271830, 4.30907338423408942578015056422, 5.41444261321502592464564147616, 6.10511133033384744873863726474, 7.916292098518255842724499864808, 8.585456689315302595364703403130, 9.517448274246607798972189506257, 10.75194394711608089947111925582, 12.34089227777744688975222980495, 13.20885075851248963350338240151

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