Properties

Label 2-12e2-144.85-c1-0-11
Degree $2$
Conductor $144$
Sign $0.960 - 0.277i$
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  + (0.268 + 1.38i)2-s + (−0.222 − 1.71i)3-s + (−1.85 + 0.746i)4-s + (2.97 − 0.798i)5-s + (2.32 − 0.771i)6-s + (1.78 − 1.02i)7-s + (−1.53 − 2.37i)8-s + (−2.90 + 0.764i)9-s + (1.90 + 3.92i)10-s + (0.119 − 0.446i)11-s + (1.69 + 3.02i)12-s + (1.52 + 5.67i)13-s + (1.90 + 2.19i)14-s + (−2.03 − 4.93i)15-s + (2.88 − 2.77i)16-s + 0.0443·17-s + ⋯
L(s)  = 1  + (0.190 + 0.981i)2-s + (−0.128 − 0.991i)3-s + (−0.927 + 0.373i)4-s + (1.33 − 0.357i)5-s + (0.949 − 0.314i)6-s + (0.673 − 0.388i)7-s + (−0.543 − 0.839i)8-s + (−0.966 + 0.254i)9-s + (0.603 + 1.24i)10-s + (0.0360 − 0.134i)11-s + (0.489 + 0.871i)12-s + (0.421 + 1.57i)13-s + (0.509 + 0.587i)14-s + (−0.525 − 1.27i)15-s + (0.721 − 0.692i)16-s + 0.0107·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.960 - 0.277i$
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: \((2,\ 144,\ (\ :1/2),\ 0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26905 + 0.179765i\)
\(L(\frac12)\) \(\approx\) \(1.26905 + 0.179765i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.268 - 1.38i)T \)
3 \( 1 + (0.222 + 1.71i)T \)
good5 \( 1 + (-2.97 + 0.798i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.78 + 1.02i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.119 + 0.446i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.52 - 5.67i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 0.0443T + 17T^{2} \)
19 \( 1 + (1.10 - 1.10i)T - 19iT^{2} \)
23 \( 1 + (7.89 + 4.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.95 + 1.86i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.542 + 0.939i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.769 + 0.769i)T + 37iT^{2} \)
41 \( 1 + (-5.77 - 3.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.96 - 11.0i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.22 + 2.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \)
59 \( 1 + (-3.59 + 0.962i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.18 + 0.318i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.48 - 5.52i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.88iT - 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 + (-3.46 - 6.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.588 + 0.157i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 5.30iT - 89T^{2} \)
97 \( 1 + (5.88 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.46360817543739110174094247585, −12.52446735512809669041059381033, −11.33442865555395116206348571720, −9.738858589353550106483929811148, −8.728820303766613544389126683116, −7.73822616741668249924861198650, −6.45709702925505172772433727637, −5.85689434333929402676912403457, −4.42992497695140544032202279782, −1.80663719716735398386525702236, 2.19879905303174637642199681426, 3.66071977570384020151297095572, 5.32579515537549039855237603955, 5.78793951835048946226064211606, 8.268876944271699975036435123606, 9.365005944820594202483625354785, 10.18039712353868879080176759948, 10.80528386448481039763882979416, 11.84105126738216160319660293730, 13.08816258252793196401380558034

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