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Properties

Label 2-12e2-9.4-c1-0-2
Degree $2$
Conductor $144$
Sign $0.993 - 0.112i$
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  + (1.18 + 1.26i)3-s + (1.68 − 2.92i)5-s + (−0.686 − 1.18i)7-s + (−0.186 + 2.99i)9-s + (0.5 + 0.866i)11-s + (−2.68 + 4.65i)13-s + (5.68 − 1.33i)15-s + 0.372·17-s − 6.37·19-s + (0.686 − 2.27i)21-s + (2.68 − 4.65i)23-s + (−3.18 − 5.51i)25-s + (−4.00 + 3.31i)27-s + (−0.686 − 1.18i)29-s + (0.313 − 0.543i)31-s + ⋯
L(s)  = 1  + (0.684 + 0.728i)3-s + (0.754 − 1.30i)5-s + (−0.259 − 0.449i)7-s + (−0.0620 + 0.998i)9-s + (0.150 + 0.261i)11-s + (−0.745 + 1.29i)13-s + (1.46 − 0.344i)15-s + 0.0902·17-s − 1.46·19-s + (0.149 − 0.496i)21-s + (0.560 − 0.970i)23-s + (−0.637 − 1.10i)25-s + (−0.769 + 0.638i)27-s + (−0.127 − 0.220i)29-s + (0.0563 − 0.0976i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.993 - 0.112i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39361 + 0.0784358i\)
\(L(\frac12)\) \(\approx\) \(1.39361 + 0.0784358i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.18 - 1.26i)T \)
good5 \( 1 + (-1.68 + 2.92i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.686 + 1.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.68 - 4.65i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
19 \( 1 + 6.37T + 19T^{2} \)
23 \( 1 + (-2.68 + 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.686 + 1.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.313 + 0.543i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.74T + 37T^{2} \)
41 \( 1 + (-0.127 + 0.221i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.87 - 8.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.686 + 1.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.68 - 2.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.87 + 6.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + (0.313 + 0.543i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.68 - 13.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-4.87 - 8.43i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.17157947551292734769979113801, −12.39032336939549875657846694695, −10.86401413960326389516098675058, −9.719821309270489066472683815633, −9.168641104253105348380164417819, −8.207484316760569130376985957768, −6.61774342285737836548211552451, −4.96380235069853970123839412052, −4.20050406750091340005308759601, −2.11686393901895015657619816444, 2.30975243910824142642966673921, 3.28763326427326427239363895032, 5.71682413458066262014292123890, 6.68090154786800399856727105782, 7.65962270926953171518328201763, 8.925412211009464225880284705913, 10.00249929753441413174352370999, 10.94051239249808914606259812776, 12.34480540869223129518117692136, 13.13872590042699410513688271596

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