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Properties

Label 2-12e2-144.59-c1-0-7
Degree $2$
Conductor $144$
Sign $0.991 + 0.130i$
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  + (−1.35 − 0.420i)2-s + (1.06 − 1.36i)3-s + (1.64 + 1.13i)4-s + (0.619 + 2.31i)5-s + (−2.01 + 1.39i)6-s + (2.51 + 4.35i)7-s + (−1.74 − 2.22i)8-s + (−0.733 − 2.90i)9-s + (0.135 − 3.38i)10-s + (−0.276 + 1.03i)11-s + (3.30 − 1.04i)12-s + (−1.07 − 4.00i)13-s + (−1.56 − 6.93i)14-s + (3.81 + 1.61i)15-s + (1.42 + 3.73i)16-s − 2.22i·17-s + ⋯
L(s)  = 1  + (−0.954 − 0.297i)2-s + (0.614 − 0.788i)3-s + (0.823 + 0.567i)4-s + (0.276 + 1.03i)5-s + (−0.821 + 0.570i)6-s + (0.949 + 1.64i)7-s + (−0.617 − 0.786i)8-s + (−0.244 − 0.969i)9-s + (0.0427 − 1.06i)10-s + (−0.0833 + 0.310i)11-s + (0.953 − 0.300i)12-s + (−0.297 − 1.11i)13-s + (−0.417 − 1.85i)14-s + (0.985 + 0.416i)15-s + (0.355 + 0.934i)16-s − 0.539i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.991 + 0.130i$
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: \((2,\ 144,\ (\ :1/2),\ 0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966885 - 0.0634908i\)
\(L(\frac12)\) \(\approx\) \(0.966885 - 0.0634908i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.420i)T \)
3 \( 1 + (-1.06 + 1.36i)T \)
good5 \( 1 + (-0.619 - 2.31i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.51 - 4.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.276 - 1.03i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.07 + 4.00i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.22iT - 17T^{2} \)
19 \( 1 + (0.697 + 0.697i)T + 19iT^{2} \)
23 \( 1 + (2.20 + 1.27i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.157 + 0.589i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.190 + 0.109i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.16 + 5.16i)T + 37iT^{2} \)
41 \( 1 + (0.828 - 1.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.98 + 1.33i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.76 - 9.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.80 + 7.80i)T - 53iT^{2} \)
59 \( 1 + (-5.09 + 1.36i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.48 + 1.73i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-8.22 + 2.20i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + (7.74 - 4.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.55 + 1.48i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 6.56T + 89T^{2} \)
97 \( 1 + (1.51 + 2.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.73712593220388287593387531628, −12.01491430869750918530343920312, −11.07718161695143346820015331162, −9.878683410144784856415203052530, −8.772628559184134140723272778721, −7.994710992814800513117185860310, −6.97524368960183996201127779449, −5.73827859107678233937249913487, −2.90432930451159611703195554143, −2.13866381973920526616751416853, 1.62087304205815672258209255168, 4.09892704796104019988454421594, 5.21613414379081118683286642319, 7.07952001033982468822418281811, 8.175441412086806193538298915143, 8.861309614659400524388334356900, 9.988634314282376290198305317370, 10.67926649966718259343205102637, 11.75032547937320065634979523469, 13.53137192666128736488196410949

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