Properties

Label 2-12e2-144.85-c1-0-0
Degree $2$
Conductor $144$
Sign $-0.947 + 0.319i$
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  + (−0.174 + 1.40i)2-s + (−1.22 + 1.21i)3-s + (−1.93 − 0.490i)4-s + (0.174 − 0.0468i)5-s + (−1.49 − 1.93i)6-s + (−4.04 + 2.33i)7-s + (1.02 − 2.63i)8-s + (0.0241 − 2.99i)9-s + (0.0351 + 0.253i)10-s + (0.160 − 0.598i)11-s + (2.98 − 1.76i)12-s + (1.18 + 4.41i)13-s + (−2.57 − 6.08i)14-s + (−0.157 + 0.270i)15-s + (3.51 + 1.90i)16-s − 4.34·17-s + ⋯
L(s)  = 1  + (−0.123 + 0.992i)2-s + (−0.709 + 0.704i)3-s + (−0.969 − 0.245i)4-s + (0.0781 − 0.0209i)5-s + (−0.611 − 0.791i)6-s + (−1.52 + 0.883i)7-s + (0.363 − 0.931i)8-s + (0.00806 − 0.999i)9-s + (0.0111 + 0.0801i)10-s + (0.0483 − 0.180i)11-s + (0.860 − 0.508i)12-s + (0.327 + 1.22i)13-s + (−0.687 − 1.62i)14-s + (−0.0407 + 0.0698i)15-s + (0.879 + 0.475i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.947 + 0.319i$
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: \((2,\ 144,\ (\ :1/2),\ -0.947 + 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0755745 - 0.460204i\)
\(L(\frac12)\) \(\approx\) \(0.0755745 - 0.460204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.174 - 1.40i)T \)
3 \( 1 + (1.22 - 1.21i)T \)
good5 \( 1 + (-0.174 + 0.0468i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (4.04 - 2.33i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.160 + 0.598i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.18 - 4.41i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 4.34T + 17T^{2} \)
19 \( 1 + (1.23 - 1.23i)T - 19iT^{2} \)
23 \( 1 + (-3.86 - 2.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.64 - 2.31i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (2.25 - 3.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.79 - 2.79i)T + 37iT^{2} \)
41 \( 1 + (3.67 + 2.12i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00351 + 0.0131i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.17 + 2.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.519 + 0.519i)T + 53iT^{2} \)
59 \( 1 + (-11.0 + 2.95i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.19 + 0.588i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.88 + 7.04i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.55iT - 71T^{2} \)
73 \( 1 + 2.92iT - 73T^{2} \)
79 \( 1 + (-1.45 - 2.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.44 + 1.99i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 3.18iT - 89T^{2} \)
97 \( 1 + (-8.03 - 13.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.71143515105215681596461048915, −12.75559564782696641990733819006, −11.69259702011046366217955798991, −10.31073085099947563691946656842, −9.302616253876564102896080854946, −8.826196294788633729962884695119, −6.72181918614921122222452638005, −6.27552163884730047444103891076, −5.03469883771249351998477621844, −3.63662934420147167770039254014, 0.52603630693713939198084320226, 2.75704272366415467626903440147, 4.34214815772166862530135264243, 5.98399942829808428311786670293, 7.09896399547310469968825463516, 8.437169036543977860930266074074, 9.903427136941297216446320265185, 10.52910266847603191287736989663, 11.50675612922256310095466685340, 12.83931699653739107081535873498

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