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Properties

Label 2-12e2-144.13-c1-0-4
Degree $2$
Conductor $144$
Sign $-0.379 - 0.925i$
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  + (1.04 + 0.948i)2-s + (−1.13 + 1.31i)3-s + (0.199 + 1.98i)4-s + (0.00302 + 0.0112i)5-s + (−2.43 + 0.300i)6-s + (−1.05 + 0.610i)7-s + (−1.67 + 2.27i)8-s + (−0.435 − 2.96i)9-s + (−0.00753 + 0.0147i)10-s + (1.83 + 0.490i)11-s + (−2.83 − 1.99i)12-s + (5.06 − 1.35i)13-s + (−1.68 − 0.362i)14-s + (−0.0182 − 0.00881i)15-s + (−3.92 + 0.795i)16-s − 1.54·17-s + ⋯
L(s)  = 1  + (0.741 + 0.670i)2-s + (−0.653 + 0.756i)3-s + (0.0999 + 0.994i)4-s + (0.00135 + 0.00504i)5-s + (−0.992 + 0.122i)6-s + (−0.399 + 0.230i)7-s + (−0.593 + 0.804i)8-s + (−0.145 − 0.989i)9-s + (−0.00238 + 0.00465i)10-s + (0.551 + 0.147i)11-s + (−0.818 − 0.574i)12-s + (1.40 − 0.376i)13-s + (−0.451 − 0.0970i)14-s + (−0.00470 − 0.00227i)15-s + (−0.980 + 0.198i)16-s − 0.374·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.379 - 0.925i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.379 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721276 + 1.07591i\)
\(L(\frac12)\) \(\approx\) \(0.721276 + 1.07591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 0.948i)T \)
3 \( 1 + (1.13 - 1.31i)T \)
good5 \( 1 + (-0.00302 - 0.0112i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.05 - 0.610i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.83 - 0.490i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-5.06 + 1.35i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 1.54T + 17T^{2} \)
19 \( 1 + (-4.06 - 4.06i)T + 19iT^{2} \)
23 \( 1 + (5.20 + 3.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.798 + 2.98i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.92 + 5.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.923 + 0.923i)T - 37iT^{2} \)
41 \( 1 + (3.20 + 1.85i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.84 + 1.29i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.31 + 2.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.88 - 8.88i)T - 53iT^{2} \)
59 \( 1 + (-2.35 - 8.78i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.24 + 12.1i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (11.8 - 3.18i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 - 4.32iT - 73T^{2} \)
79 \( 1 + (0.261 + 0.453i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.91 + 10.8i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 10.7iT - 89T^{2} \)
97 \( 1 + (-8.78 - 15.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.52145879843990760199229253939, −12.34690209515854284634536661300, −11.65401458380039595573840310815, −10.48197708543895387968390540602, −9.227494574013062126411398740612, −8.108053594063796241800326016155, −6.43602887372083957141686932922, −5.89025309558422299803301251706, −4.46335790020508748019976614957, −3.40021891608268219640152799252, 1.37937119597792766440501865919, 3.36057641070113924390476422480, 4.92729158963637660332567442475, 6.20333377200019114911762274731, 6.93582331065155197178678150889, 8.719408905625572990428553925704, 10.05648929861357108131815245791, 11.24529584389976008216714045658, 11.65806428269807100365027874102, 12.88061824634386302788117821528

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