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Properties

Label 2-12e2-144.85-c1-0-9
Degree $2$
Conductor $144$
Sign $0.737 + 0.675i$
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 + (−1.86 + 0.5i)5-s + (−1.73 − 1.73i)6-s + (3.86 − 2.23i)7-s + (2 − 2i)8-s + 2.99·9-s + (2.36 + 1.36i)10-s + (−0.5 + 1.86i)11-s + 3.46i·12-s + (−0.598 − 2.23i)13-s + (−6.09 − 1.63i)14-s + (−3.23 + 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 + (−0.834 + 0.223i)5-s + (−0.707 − 0.707i)6-s + (1.46 − 0.843i)7-s + (0.707 − 0.707i)8-s + 0.999·9-s + (0.748 + 0.431i)10-s + (−0.150 + 0.562i)11-s + 0.999i·12-s + (−0.165 − 0.619i)13-s + (−1.62 − 0.436i)14-s + (−0.834 + 0.223i)15-s − 16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.737 + 0.675i$
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: \((2,\ 144,\ (\ :1/2),\ 0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.988813 - 0.384528i\)
\(L(\frac12)\) \(\approx\) \(0.988813 - 0.384528i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 - 0.5i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.86 + 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.5 - 1.86i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.598 + 2.23i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + (5.59 + 3.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.866 + 0.232i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.59 - 7.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.26 + 4.26i)T + 37iT^{2} \)
41 \( 1 + (-0.696 - 0.401i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.69 - 6.33i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.598 + 1.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.73 - 5.73i)T + 53iT^{2} \)
59 \( 1 + (1.5 - 0.401i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.13 + 0.571i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.23 + 8.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 + 7.46iT - 73T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (14.1 + 3.79i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 15.8iT - 89T^{2} \)
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

−12.74125520720558973744745325884, −11.93393040880605976200363169760, −10.67905587455816994362645821386, −10.14025149518488798944441985790, −8.621675900022574114417552356063, −7.79202337721623209110885083841, −7.41543808765625467287415855638, −4.50250297186734947751314730599, −3.52482606450636738741377164493, −1.75626580449775310729311826213, 1.97125040787209740677938336346, 4.22905994998182250667089184265, 5.55595172985257798959387442574, 7.32426765560973569965254989170, 8.173910521797010645516523671868, 8.630019775498156883545730881250, 9.775831417875831444823646892583, 11.17322122382068901379214541177, 12.01195979020728751943857744650, 13.64340490316804689700597176859

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