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Properties

Label 2-12e2-144.85-c1-0-14
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: $\chi_{144} (85, \cdot )$
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

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

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