Properties

Label 2-12e2-144.13-c1-0-5
Degree $2$
Conductor $144$
Sign $0.177 - 0.984i$
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  + (−0.671 + 1.24i)2-s + (1.72 − 0.162i)3-s + (−1.09 − 1.67i)4-s + (0.592 + 2.21i)5-s + (−0.956 + 2.25i)6-s + (−2.67 + 1.54i)7-s + (2.81 − 0.241i)8-s + (2.94 − 0.559i)9-s + (−3.14 − 0.748i)10-s + (3.42 + 0.918i)11-s + (−2.16 − 2.70i)12-s + (1.40 − 0.375i)13-s + (−0.124 − 4.35i)14-s + (1.38 + 3.71i)15-s + (−1.59 + 3.66i)16-s − 1.69·17-s + ⋯
L(s)  = 1  + (−0.475 + 0.879i)2-s + (0.995 − 0.0936i)3-s + (−0.548 − 0.836i)4-s + (0.264 + 0.988i)5-s + (−0.390 + 0.920i)6-s + (−1.00 + 0.582i)7-s + (0.996 − 0.0852i)8-s + (0.982 − 0.186i)9-s + (−0.996 − 0.236i)10-s + (1.03 + 0.277i)11-s + (−0.624 − 0.781i)12-s + (0.389 − 0.104i)13-s + (−0.0331 − 1.16i)14-s + (0.356 + 0.959i)15-s + (−0.398 + 0.917i)16-s − 0.411·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.177 - 0.984i$
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: \((2,\ 144,\ (\ :1/2),\ 0.177 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863692 + 0.721774i\)
\(L(\frac12)\) \(\approx\) \(0.863692 + 0.721774i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.671 - 1.24i)T \)
3 \( 1 + (-1.72 + 0.162i)T \)
good5 \( 1 + (-0.592 - 2.21i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.67 - 1.54i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.42 - 0.918i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.40 + 0.375i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 1.69T + 17T^{2} \)
19 \( 1 + (5.41 + 5.41i)T + 19iT^{2} \)
23 \( 1 + (3.69 + 2.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.147 + 0.550i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-3.59 + 6.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.59 + 2.59i)T - 37iT^{2} \)
41 \( 1 + (-8.14 - 4.70i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.5 + 2.81i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.322 + 0.558i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.59 - 7.59i)T - 53iT^{2} \)
59 \( 1 + (1.55 + 5.82i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.04 + 3.88i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.02 + 1.07i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.81iT - 71T^{2} \)
73 \( 1 + 0.0254iT - 73T^{2} \)
79 \( 1 + (-7.90 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.50 - 9.35i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.29iT - 89T^{2} \)
97 \( 1 + (-4.31 - 7.46i)T + (-48.5 + 84.0i)T^{2} \)
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

−13.61471135460594049323476330854, −12.71494778950219406582122366989, −10.96013690671478601755999690867, −9.780355265875620947001349221671, −9.185107455875748820604708821863, −8.133906781306344398324196773834, −6.63103173424708491849577937748, −6.43968322223574698707253312758, −4.17412272119277732118798468384, −2.47802810580879995628389108282, 1.56013491937843888351242037489, 3.45295181548362768904442104336, 4.34766978269543131371731084798, 6.57914357881792951950614812912, 8.128694726836196599087043888188, 8.903005234970544550539242645627, 9.675949083009300676448065860803, 10.52896302807167478283185197020, 12.06675875174194226083511070897, 12.92577110311065039101437131082

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