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Properties

Label 2-12e2-144.13-c1-0-6
Degree $2$
Conductor $144$
Sign $0.999 - 0.0436i$
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 + i)2-s − 1.73·3-s − 2i·4-s + (−0.133 − 0.5i)5-s + (1.73 − 1.73i)6-s + (2.13 − 1.23i)7-s + (2 + 2i)8-s + 2.99·9-s + (0.633 + 0.366i)10-s + (−0.5 − 0.133i)11-s + 3.46i·12-s + (4.59 − 1.23i)13-s + (−0.901 + 3.36i)14-s + (0.232 + 0.866i)15-s − 4·16-s + 4·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00·3-s − i·4-s + (−0.0599 − 0.223i)5-s + (0.707 − 0.707i)6-s + (0.806 − 0.465i)7-s + (0.707 + 0.707i)8-s + 0.999·9-s + (0.200 + 0.115i)10-s + (−0.150 − 0.0403i)11-s + 0.999i·12-s + (1.27 − 0.341i)13-s + (−0.241 + 0.899i)14-s + (0.0599 + 0.223i)15-s − 16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.999 - 0.0436i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.999 - 0.0436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.648827 + 0.0141574i\)
\(L(\frac12)\) \(\approx\) \(0.648827 + 0.0141574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + 1.73T \)
good5 \( 1 + (0.133 + 0.5i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.13 + 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.133i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.59 + 1.23i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + (0.401 + 0.232i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.866 + 3.23i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.598 + 1.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.73 - 7.73i)T - 37iT^{2} \)
41 \( 1 + (9.69 + 5.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.69 - 2.33i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.59 - 7.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.26 + 2.26i)T - 53iT^{2} \)
59 \( 1 + (1.5 + 5.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.86 - 14.4i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.23 + 0.330i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 0.535iT - 73T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.16 + 11.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
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

−13.19026050877798403199643942712, −11.82811333555964085713992995463, −10.81228816357909975190494939729, −10.27407755158786029840891151149, −8.763488774264092612253452501915, −7.79962256771825930189388813332, −6.64156703826392183683248293265, −5.57501254924254535274991729191, −4.47712495211848914475991319712, −1.13575259775052458589247932869, 1.58469767340781759911411983391, 3.75466490335018808154778992852, 5.29689421729481690546994889802, 6.72715406587952014398251117116, 7.991003120570099276850228682624, 9.027960482057989728412776109877, 10.45055405139216194905977153022, 10.93241318877866781640910256722, 11.93704074158928682994842540721, 12.57653125760626384808235654671

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