Properties

Label 2-12e2-144.13-c1-0-19
Degree $2$
Conductor $144$
Sign $-0.142 + 0.989i$
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  + (1.26 − 0.634i)2-s + (−1.68 − 0.409i)3-s + (1.19 − 1.60i)4-s + (−0.884 − 3.30i)5-s + (−2.38 + 0.550i)6-s + (−2.63 + 1.51i)7-s + (0.492 − 2.78i)8-s + (2.66 + 1.37i)9-s + (−3.21 − 3.61i)10-s + (5.21 + 1.39i)11-s + (−2.66 + 2.21i)12-s + (1.41 − 0.378i)13-s + (−2.36 + 3.59i)14-s + (0.137 + 5.91i)15-s + (−1.14 − 3.83i)16-s + 0.259·17-s + ⋯
L(s)  = 1  + (0.893 − 0.448i)2-s + (−0.971 − 0.236i)3-s + (0.597 − 0.801i)4-s + (−0.395 − 1.47i)5-s + (−0.974 + 0.224i)6-s + (−0.994 + 0.574i)7-s + (0.174 − 0.984i)8-s + (0.888 + 0.459i)9-s + (−1.01 − 1.14i)10-s + (1.57 + 0.421i)11-s + (−0.770 + 0.638i)12-s + (0.391 − 0.104i)13-s + (−0.631 + 0.959i)14-s + (0.0355 + 1.52i)15-s + (−0.286 − 0.958i)16-s + 0.0629·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.142 + 0.989i$
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: \((2,\ 144,\ (\ :1/2),\ -0.142 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828131 - 0.956178i\)
\(L(\frac12)\) \(\approx\) \(0.828131 - 0.956178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.634i)T \)
3 \( 1 + (1.68 + 0.409i)T \)
good5 \( 1 + (0.884 + 3.30i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.63 - 1.51i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.21 - 1.39i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.41 + 0.378i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 0.259T + 17T^{2} \)
19 \( 1 + (-0.228 - 0.228i)T + 19iT^{2} \)
23 \( 1 + (-2.69 - 1.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.438 - 1.63i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (3.30 - 5.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.24 + 1.24i)T - 37iT^{2} \)
41 \( 1 + (-8.85 - 5.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.69 + 0.722i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.24 + 1.24i)T - 53iT^{2} \)
59 \( 1 + (0.194 + 0.725i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.16 - 4.36i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.53 - 0.411i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.68iT - 71T^{2} \)
73 \( 1 - 15.1iT - 73T^{2} \)
79 \( 1 + (0.738 + 1.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.908 + 3.39i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 15.7iT - 89T^{2} \)
97 \( 1 + (-5.94 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.65857395893966980385797326273, −12.06713694186431644466112560318, −11.32338567310763913600812477948, −9.837103783661345207341668869416, −8.942249565190974619460985377898, −6.99752697318267631533076997839, −5.98729834552806919277569726556, −4.95588489344748231486191837666, −3.82548130433154396713646785740, −1.29148349040157616874845425160, 3.35074161569206822159831767250, 4.15325340048738790018261232354, 6.12537240425574922969425486576, 6.56084324804798735812351231088, 7.43725683222946802310461402165, 9.462140742839828089196037257538, 10.83984173676006569152332309628, 11.31430993573813469204095341986, 12.32316217482955319724905097826, 13.44597239827550125624334989076

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