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Properties

Label 2-12e2-144.59-c1-0-20
Degree $2$
Conductor $144$
Sign $-0.705 + 0.708i$
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  + (0.714 − 1.22i)2-s + (−1.67 + 0.433i)3-s + (−0.979 − 1.74i)4-s + (−0.247 − 0.923i)5-s + (−0.669 + 2.35i)6-s + (−1.93 − 3.35i)7-s + (−2.82 − 0.0505i)8-s + (2.62 − 1.45i)9-s + (−1.30 − 0.357i)10-s + (0.936 − 3.49i)11-s + (2.39 + 2.49i)12-s + (1.72 + 6.43i)13-s + (−5.47 − 0.0325i)14-s + (0.815 + 1.44i)15-s + (−2.08 + 3.41i)16-s − 3.74i·17-s + ⋯
L(s)  = 1  + (0.505 − 0.863i)2-s + (−0.968 + 0.250i)3-s + (−0.489 − 0.871i)4-s + (−0.110 − 0.412i)5-s + (−0.273 + 0.961i)6-s + (−0.731 − 1.26i)7-s + (−0.999 − 0.0178i)8-s + (0.874 − 0.484i)9-s + (−0.412 − 0.113i)10-s + (0.282 − 1.05i)11-s + (0.692 + 0.721i)12-s + (0.478 + 1.78i)13-s + (−1.46 − 0.00870i)14-s + (0.210 + 0.372i)15-s + (−0.520 + 0.853i)16-s − 0.907i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.705 + 0.708i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.705 + 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.338665 - 0.815342i\)
\(L(\frac12)\) \(\approx\) \(0.338665 - 0.815342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.714 + 1.22i)T \)
3 \( 1 + (1.67 - 0.433i)T \)
good5 \( 1 + (0.247 + 0.923i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.93 + 3.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.936 + 3.49i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.72 - 6.43i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 3.74iT - 17T^{2} \)
19 \( 1 + (-3.09 - 3.09i)T + 19iT^{2} \)
23 \( 1 + (-0.327 - 0.188i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.10 + 4.14i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.788 - 0.455i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.13 + 2.13i)T + 37iT^{2} \)
41 \( 1 + (-3.66 + 6.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.47 - 0.662i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.0726 + 0.125i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.67 - 5.67i)T - 53iT^{2} \)
59 \( 1 + (-3.99 + 1.07i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.33 + 1.69i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.33 + 0.357i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.88iT - 71T^{2} \)
73 \( 1 - 6.65iT - 73T^{2} \)
79 \( 1 + (2.18 - 1.26i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.261 + 0.0699i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 5.86T + 89T^{2} \)
97 \( 1 + (-5.07 - 8.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.53426229522746467968997417403, −11.62911842336959011262837933137, −10.95056628105202562655718639726, −9.911489119443328170858324745200, −9.052957505699245989185494481005, −6.94762634335531812835502384931, −5.96040638734775343733491374518, −4.51674825375158123233996882301, −3.67004754726796160741796264652, −0.917052504401855907866085446344, 3.12339169079669939478951402342, 4.97359610820516348539130930672, 5.90430709026713596466213043025, 6.76028887588468352088453787411, 7.891428947986339259549246642873, 9.249929156442677113165578241840, 10.51525359283383454843377932039, 11.83826164004201213171052238493, 12.70372457090812672189801042841, 13.08041498100941834248130918210

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