Properties

Label 2-12e2-144.11-c1-0-20
Degree $2$
Conductor $144$
Sign $-0.305 + 0.952i$
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  + (1.21 − 0.716i)2-s + (−1.58 − 0.692i)3-s + (0.972 − 1.74i)4-s + (−2.83 − 0.759i)5-s + (−2.43 + 0.293i)6-s + (1.41 − 2.45i)7-s + (−0.0675 − 2.82i)8-s + (2.04 + 2.19i)9-s + (−3.99 + 1.10i)10-s + (0.794 − 0.212i)11-s + (−2.75 + 2.10i)12-s + (3.22 + 0.864i)13-s + (−0.0319 − 4.00i)14-s + (3.97 + 3.16i)15-s + (−2.10 − 3.39i)16-s + 7.28i·17-s + ⋯
L(s)  = 1  + (0.862 − 0.506i)2-s + (−0.916 − 0.399i)3-s + (0.486 − 0.873i)4-s + (−1.26 − 0.339i)5-s + (−0.992 + 0.120i)6-s + (0.535 − 0.927i)7-s + (−0.0238 − 0.999i)8-s + (0.680 + 0.732i)9-s + (−1.26 + 0.349i)10-s + (0.239 − 0.0641i)11-s + (−0.794 + 0.606i)12-s + (0.894 + 0.239i)13-s + (−0.00853 − 1.07i)14-s + (1.02 + 0.817i)15-s + (−0.527 − 0.849i)16-s + 1.76i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.305 + 0.952i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.305 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702843 - 0.963852i\)
\(L(\frac12)\) \(\approx\) \(0.702843 - 0.963852i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 + 0.716i)T \)
3 \( 1 + (1.58 + 0.692i)T \)
good5 \( 1 + (2.83 + 0.759i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.41 + 2.45i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.794 + 0.212i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.22 - 0.864i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 7.28iT - 17T^{2} \)
19 \( 1 + (0.951 + 0.951i)T + 19iT^{2} \)
23 \( 1 + (-5.13 + 2.96i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.76 - 0.473i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.05 + 1.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.03 - 6.03i)T + 37iT^{2} \)
41 \( 1 + (4.60 + 7.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.402 - 1.50i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.85 - 4.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.50 + 4.50i)T - 53iT^{2} \)
59 \( 1 + (2.85 - 10.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.16 + 8.08i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.99 - 11.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.98iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (8.94 + 5.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.00 + 3.74i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 9.25T + 89T^{2} \)
97 \( 1 + (-0.148 + 0.257i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.74394071742996324601309035321, −11.74739897960776526135157965010, −11.07497090094505348337518897758, −10.43063841342168950765952841213, −8.387920015143477352769568526181, −7.19525041472814269935977893697, −6.12324569019039964516573028335, −4.60329634216395293752443962181, −3.90454109908047723783831371410, −1.20320517485560631221117053512, 3.30007200009331022370235197300, 4.57249920575432824232044656270, 5.55251034370061160585349962946, 6.80711752409630071209787872533, 7.82753356087554492612481145290, 9.106703025103152785079802830101, 11.00989857941790906540046033738, 11.57417665911890983453745573904, 12.10794310153254671284042044332, 13.33028085941068587007803197454

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