Properties

Label 2-12e2-144.85-c1-0-12
Degree $2$
Conductor $144$
Sign $0.953 - 0.300i$
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  + (1.36 − 0.366i)2-s + 1.73i·3-s + (1.73 − i)4-s + (0.5 − 0.133i)5-s + (0.633 + 2.36i)6-s + (−2.13 + 1.23i)7-s + (1.99 − 2i)8-s − 2.99·9-s + (0.633 − 0.366i)10-s + (0.133 − 0.5i)11-s + (1.73 + 2.99i)12-s + (−1.23 − 4.59i)13-s + (−2.46 + 2.46i)14-s + (0.232 + 0.866i)15-s + (1.99 − 3.46i)16-s + 4·17-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + 0.999i·3-s + (0.866 − 0.5i)4-s + (0.223 − 0.0599i)5-s + (0.258 + 0.965i)6-s + (−0.806 + 0.465i)7-s + (0.707 − 0.707i)8-s − 0.999·9-s + (0.200 − 0.115i)10-s + (0.0403 − 0.150i)11-s + (0.499 + 0.866i)12-s + (−0.341 − 1.27i)13-s + (−0.658 + 0.658i)14-s + (0.0599 + 0.223i)15-s + (0.499 − 0.866i)16-s + 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.953 - 0.300i$
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: \((2,\ 144,\ (\ :1/2),\ 0.953 - 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75241 + 0.269722i\)
\(L(\frac12)\) \(\approx\) \(1.75241 + 0.269722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 - 1.73iT \)
good5 \( 1 + (-0.5 + 0.133i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.13 - 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.133 + 0.5i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.23 + 4.59i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + (-0.401 - 0.232i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.23 + 0.866i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.598 + 1.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.73 + 7.73i)T + 37iT^{2} \)
41 \( 1 + (-9.69 - 5.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.33 - 8.69i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.59 - 7.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.26 - 2.26i)T + 53iT^{2} \)
59 \( 1 + (-5.59 + 1.5i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-14.4 - 3.86i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.330 + 1.23i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + 0.535iT - 73T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.7 + 3.16i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 11.8iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.01648455786503649345088323325, −12.32379178754206470319813735387, −11.12229999866517874125279465521, −10.17270517682600001616098042965, −9.454431403307577527011561784855, −7.82389321818402387227233563953, −6.01141520491115696719905698434, −5.43895307090874604214781066742, −3.88642190570248213671668037780, −2.81411020733972358810975722088, 2.22104297567126918340592383475, 3.80260068693803941041898780837, 5.47860396950598096549537052584, 6.70449236918892132066392211244, 7.17584423255542462323549183039, 8.616955256663735367176161968425, 10.14534659160905082227536938475, 11.51524755483478561877894215069, 12.26880261816401330269088579088, 13.16440521696243116507826904271

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