Properties

Label 2-12e2-144.83-c1-0-9
Degree $2$
Conductor $144$
Sign $0.229 - 0.973i$
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  + (1.19 + 0.753i)2-s + (−0.256 + 1.71i)3-s + (0.864 + 1.80i)4-s + (0.546 − 2.03i)5-s + (−1.59 + 1.85i)6-s + (−0.0638 + 0.110i)7-s + (−0.323 + 2.80i)8-s + (−2.86 − 0.878i)9-s + (2.18 − 2.02i)10-s + (−0.181 − 0.678i)11-s + (−3.31 + 1.01i)12-s + (0.493 − 1.84i)13-s + (−0.159 + 0.0842i)14-s + (3.35 + 1.45i)15-s + (−2.50 + 3.11i)16-s − 4.32i·17-s + ⋯
L(s)  = 1  + (0.846 + 0.532i)2-s + (−0.148 + 0.988i)3-s + (0.432 + 0.901i)4-s + (0.244 − 0.911i)5-s + (−0.652 + 0.758i)6-s + (−0.0241 + 0.0417i)7-s + (−0.114 + 0.993i)8-s + (−0.956 − 0.292i)9-s + (0.692 − 0.641i)10-s + (−0.0548 − 0.204i)11-s + (−0.955 + 0.294i)12-s + (0.136 − 0.511i)13-s + (−0.0426 + 0.0225i)14-s + (0.865 + 0.376i)15-s + (−0.626 + 0.779i)16-s − 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27534 + 1.00933i\)
\(L(\frac12)\) \(\approx\) \(1.27534 + 1.00933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 0.753i)T \)
3 \( 1 + (0.256 - 1.71i)T \)
good5 \( 1 + (-0.546 + 2.03i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.0638 - 0.110i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.181 + 0.678i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.493 + 1.84i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 4.32iT - 17T^{2} \)
19 \( 1 + (3.97 - 3.97i)T - 19iT^{2} \)
23 \( 1 + (-6.81 + 3.93i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.248 + 0.926i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.91 + 2.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.64 - 6.64i)T - 37iT^{2} \)
41 \( 1 + (4.61 + 7.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.8 - 2.91i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.92 - 10.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.00259 + 0.00259i)T + 53iT^{2} \)
59 \( 1 + (4.09 + 1.09i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.44 + 1.19i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.01 - 0.538i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.80iT - 71T^{2} \)
73 \( 1 - 1.87iT - 73T^{2} \)
79 \( 1 + (3.00 + 1.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.47 - 0.394i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (-3.31 + 5.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.36524294346283156690941928698, −12.46163448427747889612432322544, −11.47041367910595952627407763090, −10.33288155227994102956041254775, −8.988773048165034855621849799264, −8.210297839359428663021517691695, −6.49750906068833414820967006066, −5.29207431240547670970814702688, −4.58997029917843583484450164522, −3.10687889172965731451875486703, 1.91603226137968578877721422228, 3.27591576951894572135315323147, 5.11015621640193630449622018784, 6.53200774079674999909323596124, 6.94229512228962828189508255949, 8.701809327872264378959373030690, 10.29332860145090794253369950318, 11.08022326026975401284502299676, 11.94256172397155728644358332138, 13.05281530924409948001912412623

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