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Properties

Label 2-12e2-144.61-c1-0-15
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: $\chi_{144} (61, \cdot )$
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

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

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