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Properties

Label 2-12e2-144.133-c1-0-5
Degree $2$
Conductor $144$
Sign $0.573 - 0.819i$
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  + (−1.39 − 0.237i)2-s + (1.28 + 1.15i)3-s + (1.88 + 0.662i)4-s + (−0.326 + 1.21i)5-s + (−1.52 − 1.91i)6-s + (0.707 + 0.408i)7-s + (−2.47 − 1.37i)8-s + (0.324 + 2.98i)9-s + (0.743 − 1.61i)10-s + (−1.85 + 0.497i)11-s + (1.66 + 3.03i)12-s + (−0.434 − 0.116i)13-s + (−0.889 − 0.737i)14-s + (−1.82 + 1.19i)15-s + (3.12 + 2.50i)16-s + 6.62·17-s + ⋯
L(s)  = 1  + (−0.985 − 0.168i)2-s + (0.744 + 0.667i)3-s + (0.943 + 0.331i)4-s + (−0.145 + 0.544i)5-s + (−0.621 − 0.783i)6-s + (0.267 + 0.154i)7-s + (−0.874 − 0.485i)8-s + (0.108 + 0.994i)9-s + (0.235 − 0.511i)10-s + (−0.559 + 0.149i)11-s + (0.481 + 0.876i)12-s + (−0.120 − 0.0322i)13-s + (−0.237 − 0.197i)14-s + (−0.471 + 0.307i)15-s + (0.780 + 0.625i)16-s + 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.573 - 0.819i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.573 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804565 + 0.418920i\)
\(L(\frac12)\) \(\approx\) \(0.804565 + 0.418920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.237i)T \)
3 \( 1 + (-1.28 - 1.15i)T \)
good5 \( 1 + (0.326 - 1.21i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.707 - 0.408i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.85 - 0.497i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.434 + 0.116i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.62T + 17T^{2} \)
19 \( 1 + (-1.18 + 1.18i)T - 19iT^{2} \)
23 \( 1 + (2.66 - 1.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.31 + 8.65i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.61 + 7.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.14 + 2.14i)T + 37iT^{2} \)
41 \( 1 + (-9.15 + 5.28i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.19 - 1.66i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.140 - 0.244i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.83 - 4.83i)T + 53iT^{2} \)
59 \( 1 + (-1.91 + 7.15i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.64 - 9.87i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (5.22 + 1.39i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.27iT - 71T^{2} \)
73 \( 1 + 4.92iT - 73T^{2} \)
79 \( 1 + (-7.70 + 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.92 - 10.9i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 3.44iT - 89T^{2} \)
97 \( 1 + (4.46 - 7.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.27668578784496567897147536747, −11.91699894294288283067912622755, −10.92275552881494681383955465377, −10.03677513517366181482933784961, −9.298416690224077447060581672222, −7.979289616464276099577402178292, −7.45966812598132127046224008845, −5.61603367414487017500895718148, −3.66211460799751584195130945973, −2.37935441028735065629684248108, 1.36410755014570598961221951723, 3.15578726379140259289350394150, 5.42613832820225367738232455265, 6.92945285443144527706996302793, 7.88683675325544865119208801926, 8.566257672250104725884053453694, 9.632190014631508444493874069994, 10.70061756621475315137925620551, 12.09209091939591671249161856021, 12.69922656440632492919651408985

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