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Properties

Label 2-12e2-48.5-c2-0-3
Degree $2$
Conductor $144$
Sign $0.745 - 0.666i$
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  + (−0.252 − 1.98i)2-s + (−3.87 + 1.00i)4-s + (−1.66 + 1.66i)5-s + 13.3i·7-s + (2.96 + 7.42i)8-s + (3.72 + 2.88i)10-s + (7.81 − 7.81i)11-s + (−10.4 + 10.4i)13-s + (26.4 − 3.36i)14-s + (13.9 − 7.76i)16-s + 10.0i·17-s + (5.07 − 5.07i)19-s + (4.77 − 8.11i)20-s + (−17.4 − 13.5i)22-s − 29.7·23-s + ⋯
L(s)  = 1  + (−0.126 − 0.991i)2-s + (−0.968 + 0.250i)4-s + (−0.332 + 0.332i)5-s + 1.90i·7-s + (0.371 + 0.928i)8-s + (0.372 + 0.288i)10-s + (0.710 − 0.710i)11-s + (−0.807 + 0.807i)13-s + (1.88 − 0.240i)14-s + (0.874 − 0.485i)16-s + 0.590i·17-s + (0.266 − 0.266i)19-s + (0.238 − 0.405i)20-s + (−0.794 − 0.615i)22-s − 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.745 - 0.666i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.745 - 0.666i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.879100 + 0.335830i\)
\(L(\frac12)\) \(\approx\) \(0.879100 + 0.335830i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.252 + 1.98i)T \)
3 \( 1 \)
good5 \( 1 + (1.66 - 1.66i)T - 25iT^{2} \)
7 \( 1 - 13.3iT - 49T^{2} \)
11 \( 1 + (-7.81 + 7.81i)T - 121iT^{2} \)
13 \( 1 + (10.4 - 10.4i)T - 169iT^{2} \)
17 \( 1 - 10.0iT - 289T^{2} \)
19 \( 1 + (-5.07 + 5.07i)T - 361iT^{2} \)
23 \( 1 + 29.7T + 529T^{2} \)
29 \( 1 + (-18.6 - 18.6i)T + 841iT^{2} \)
31 \( 1 - 27.1T + 961T^{2} \)
37 \( 1 + (-13.3 - 13.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 34.9T + 1.68e3T^{2} \)
43 \( 1 + (-7.29 - 7.29i)T + 1.84e3iT^{2} \)
47 \( 1 + 51.5iT - 2.20e3T^{2} \)
53 \( 1 + (68.4 - 68.4i)T - 2.80e3iT^{2} \)
59 \( 1 + (-31.5 + 31.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (-72.6 + 72.6i)T - 3.72e3iT^{2} \)
67 \( 1 + (-60.3 + 60.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 3.09T + 5.04e3T^{2} \)
73 \( 1 - 2.05iT - 5.32e3T^{2} \)
79 \( 1 - 53.3T + 6.24e3T^{2} \)
83 \( 1 + (-21.7 - 21.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 137.T + 7.92e3T^{2} \)
97 \( 1 + 17.1T + 9.40e3T^{2} \)
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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

−12.53019947697376553488125274644, −11.87789655793972786411442940689, −11.29958320209649126308063723092, −9.839364768776599550920295215804, −8.986828018544563426272813350147, −8.155933569765244101588706712890, −6.30980945442175983228178830444, −4.98434252865566319073971126321, −3.38861680967835851646868760988, −2.08242674660272292524648784527, 0.66815775141156800920055142750, 3.96742449234701006881311338908, 4.78131781877607982770782101662, 6.49203116402378720799477300068, 7.46053004924692275248212779330, 8.135936783808576512459961716243, 9.801770553452201379969243575405, 10.21640014011392744198724198217, 11.89723017107907194536225729741, 13.00449208748338386192799519379

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