Properties

Label 2-12e2-144.61-c1-0-11
Degree $2$
Conductor $144$
Sign $0.737 - 0.675i$
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  + (0.366 + 1.36i)2-s + (0.866 − 1.5i)3-s + (−1.73 + i)4-s + (1 + 0.267i)5-s + (2.36 + 0.633i)6-s + (2.36 + 1.36i)7-s + (−2 − 1.99i)8-s + (−1.5 − 2.59i)9-s + 1.46i·10-s + (1.13 + 4.23i)11-s + 3.46i·12-s + (0.901 − 3.36i)13-s + (−0.999 + 3.73i)14-s + (1.26 − 1.26i)15-s + (1.99 − 3.46i)16-s − 5.73·17-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (0.499 − 0.866i)3-s + (−0.866 + 0.5i)4-s + (0.447 + 0.119i)5-s + (0.965 + 0.258i)6-s + (0.894 + 0.516i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)9-s + 0.462i·10-s + (0.341 + 1.27i)11-s + 0.999i·12-s + (0.250 − 0.933i)13-s + (−0.267 + 0.997i)14-s + (0.327 − 0.327i)15-s + (0.499 − 0.866i)16-s − 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.737 - 0.675i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.737 - 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33263 + 0.518231i\)
\(L(\frac12)\) \(\approx\) \(1.33263 + 0.518231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 + (-0.866 + 1.5i)T \)
good5 \( 1 + (-1 - 0.267i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.36 - 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.13 - 4.23i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.901 + 3.36i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + (2.36 + 2.36i)T + 19iT^{2} \)
23 \( 1 + (4.09 - 2.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.36 + 0.633i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.267 + 0.464i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.73 + 4.73i)T - 37iT^{2} \)
41 \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.23 - 8.33i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.83 + 6.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.46 - 7.46i)T - 53iT^{2} \)
59 \( 1 + (-7.33 - 1.96i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-11.1 + 3i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.76 - 6.59i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 + 6.26iT - 73T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.36 - 0.366i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.32083695089443945788203357704, −12.61637220742050504617444878589, −11.53559623076194766982209710882, −9.756896021186314798326279182382, −8.690942904981951103212933914153, −7.86991057058215315590972851660, −6.81034816487123612174399055683, −5.78395227319685843449626916722, −4.34546274203470849940350431922, −2.26569361244856301475568674860, 2.03927772605864394853899138606, 3.79695875238478591206756505681, 4.65353205644240147684521217007, 6.06638622518396496763135835197, 8.317960413009994173860736631235, 8.949463079538421767894310194195, 10.08277951912110871675076194763, 11.02287539217196225562641903973, 11.57381781142887454534009913713, 13.23596127546621509435153739730

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