↑

Properties

Label 2-12e2-144.85-c1-0-20
Degree $2$
Conductor $144$
Sign $-0.624 - 0.780i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.812 − 1.15i)2-s + (−0.460 − 1.66i)3-s + (−0.678 + 1.88i)4-s + (−2.69 + 0.722i)5-s + (−1.55 + 1.89i)6-s + (−2.89 + 1.67i)7-s + (2.72 − 0.743i)8-s + (−2.57 + 1.53i)9-s + (3.02 + 2.53i)10-s + (1.23 − 4.60i)11-s + (3.45 + 0.267i)12-s + (0.398 + 1.48i)13-s + (4.28 + 1.99i)14-s + (2.44 + 4.17i)15-s + (−3.07 − 2.55i)16-s − 6.47·17-s + ⋯
L(s)  = 1  + (−0.574 − 0.818i)2-s + (−0.265 − 0.963i)3-s + (−0.339 + 0.940i)4-s + (−1.20 + 0.323i)5-s + (−0.636 + 0.771i)6-s + (−1.09 + 0.631i)7-s + (0.964 − 0.262i)8-s + (−0.858 + 0.512i)9-s + (0.957 + 0.801i)10-s + (0.371 − 1.38i)11-s + (0.997 + 0.0770i)12-s + (0.110 + 0.412i)13-s + (1.14 + 0.532i)14-s + (0.632 + 1.07i)15-s + (−0.769 − 0.638i)16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.624 - 0.780i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.624 - 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0516391 + 0.107433i\)
\(L(\frac12)\) \(\approx\) \(0.0516391 + 0.107433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.812 + 1.15i)T \)
3 \( 1 + (0.460 + 1.66i)T \)
good5 \( 1 + (2.69 - 0.722i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.89 - 1.67i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.23 + 4.60i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.398 - 1.48i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 + (-0.957 + 0.957i)T - 19iT^{2} \)
23 \( 1 + (3.70 + 2.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.07 + 0.289i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-1.89 + 3.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.14 + 6.14i)T + 37iT^{2} \)
41 \( 1 + (-5.04 - 2.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.69 + 6.31i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.81 - 3.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.762 - 0.762i)T + 53iT^{2} \)
59 \( 1 + (8.11 - 2.17i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.74 - 1.80i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.487 + 1.81i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.98iT - 71T^{2} \)
73 \( 1 - 5.45iT - 73T^{2} \)
79 \( 1 + (2.95 + 5.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 + 2.77i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.24iT - 89T^{2} \)
97 \( 1 + (-6.50 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.20510086711206595364504072268, −11.49011355637111527554873158329, −10.84613062030918163142084035963, −9.113741279683952269835092007905, −8.408281632768197325191359801578, −7.21767978071445166377684939105, −6.17494675585012611991566273592, −3.88997726536935051191238968038, −2.62088580715596854443913883261, −0.14197295850778467913460756065, 3.88726864658930727271560485414, 4.74776106742546243660140744156, 6.39197465681508658166581042407, 7.37943220394951268363421215031, 8.627093913473131111210703478412, 9.636170950423455545508023706517, 10.37101038448031822658635541714, 11.51957984758111835535105178180, 12.70977772664836782037506827860, 14.03047550686986738940331725181

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