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Properties

Label 2-12e2-144.11-c1-0-18
Degree $2$
Conductor $144$
Sign $0.840 + 0.542i$
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  + (1.41 + 0.00841i)2-s + (0.433 − 1.67i)3-s + (1.99 + 0.0238i)4-s + (−0.923 − 0.247i)5-s + (0.626 − 2.36i)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 + (3.49 − 0.936i)11-s + (0.906 − 3.34i)12-s + (−6.43 − 1.72i)13-s + (−2.76 + 4.72i)14-s + (−0.815 + 1.44i)15-s + (3.99 + 0.0952i)16-s + 3.74i·17-s + ⋯
L(s)  = 1  + (0.999 + 0.00595i)2-s + (0.250 − 0.968i)3-s + (0.999 + 0.0119i)4-s + (−0.412 − 0.110i)5-s + (0.255 − 0.966i)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 + (1.05 − 0.282i)11-s + (0.261 − 0.965i)12-s + (−1.78 − 0.478i)13-s + (−0.738 + 1.26i)14-s + (−0.210 + 0.372i)15-s + (0.999 + 0.0238i)16-s + 0.907i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.840 + 0.542i$
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: \((2,\ 144,\ (\ :1/2),\ 0.840 + 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75799 - 0.518099i\)
\(L(\frac12)\) \(\approx\) \(1.75799 - 0.518099i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.00841i)T \)
3 \( 1 + (-0.433 + 1.67i)T \)
good5 \( 1 + (0.923 + 0.247i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.93 - 3.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.49 + 0.936i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (6.43 + 1.72i)T + (11.2 + 6.5i)T^{2} \)
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 \( 1 + (-4.14 + 1.10i)T + (25.1 - 14.5i)T^{2} \)
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 \( 1 + (0.662 + 2.47i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.0726 + 0.125i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.67 + 5.67i)T - 53iT^{2} \)
59 \( 1 + (-1.07 + 3.99i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.69 - 6.33i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.357 - 1.33i)T + (-58.0 - 33.5i)T^{2} \)
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 \( 1 + (0.0699 + 0.261i)T + (-71.8 + 41.5i)T^{2} \)
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

−12.75326873637104512778888746260, −12.14484298794987463325406647397, −11.79146594656468872268583413516, −9.974227419163343870319329845625, −8.605478880628067951413562102636, −7.44961793744163825174272902172, −6.36918274034460902919150239078, −5.44242662552622170306202812500, −3.54985787008262821516133258691, −2.27121385717602140832234663856, 2.98881969431127000975737198929, 4.12027293857134684252734150689, 4.95629935966812635249980297720, 6.78005337270467199983878042160, 7.49504973152218392545313004815, 9.476454245813333562704377397360, 10.11628581802631797871188330242, 11.38367301333683707760329630348, 12.05998174597000711956986996233, 13.50384150978027793117886748436

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