Properties

Label 2-12e2-144.85-c1-0-13
Degree $2$
Conductor $144$
Sign $0.565 + 0.825i$
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  + (−1.38 + 0.276i)2-s + (0.841 − 1.51i)3-s + (1.84 − 0.765i)4-s + (−0.0691 + 0.0185i)5-s + (−0.748 + 2.33i)6-s + (1.28 − 0.740i)7-s + (−2.35 + 1.57i)8-s + (−1.58 − 2.54i)9-s + (0.0907 − 0.0447i)10-s + (0.587 − 2.19i)11-s + (0.394 − 3.44i)12-s + (0.104 + 0.388i)13-s + (−1.57 + 1.38i)14-s + (−0.0301 + 0.120i)15-s + (2.82 − 2.82i)16-s − 0.851·17-s + ⋯
L(s)  = 1  + (−0.980 + 0.195i)2-s + (0.485 − 0.874i)3-s + (0.923 − 0.382i)4-s + (−0.0309 + 0.00828i)5-s + (−0.305 + 0.952i)6-s + (0.484 − 0.279i)7-s + (−0.831 + 0.555i)8-s + (−0.528 − 0.849i)9-s + (0.0287 − 0.0141i)10-s + (0.177 − 0.660i)11-s + (0.113 − 0.993i)12-s + (0.0288 + 0.107i)13-s + (−0.420 + 0.368i)14-s + (−0.00777 + 0.0310i)15-s + (0.706 − 0.707i)16-s − 0.206·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.565 + 0.825i$
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: \((2,\ 144,\ (\ :1/2),\ 0.565 + 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.769559 - 0.405664i\)
\(L(\frac12)\) \(\approx\) \(0.769559 - 0.405664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.276i)T \)
3 \( 1 + (-0.841 + 1.51i)T \)
good5 \( 1 + (0.0691 - 0.0185i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.28 + 0.740i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.587 + 2.19i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.104 - 0.388i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 0.851T + 17T^{2} \)
19 \( 1 + (-3.75 + 3.75i)T - 19iT^{2} \)
23 \( 1 + (-7.44 - 4.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.77 + 1.27i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.50 - 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.13 - 4.13i)T + 37iT^{2} \)
41 \( 1 + (2.05 + 1.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.669 - 2.49i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.42 - 5.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.95 - 3.95i)T + 53iT^{2} \)
59 \( 1 + (-13.7 + 3.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.28 + 0.881i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.73 + 6.47i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.362iT - 71T^{2} \)
73 \( 1 - 15.8iT - 73T^{2} \)
79 \( 1 + (5.45 + 9.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.37 + 1.43i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 + (0.627 + 1.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.02115151745883375473664702244, −11.58344522858692429959925486728, −11.11147258989697104980143692516, −9.480738209985088133467840347302, −8.750043178950121797052144368179, −7.62692955689352656191690226028, −6.94404611960169923298370446755, −5.56269953140983596980129142283, −3.10546764464055610225391452869, −1.33639260424059596164898803919, 2.25867597948299267514705370163, 3.86569796192806659435943000336, 5.51109464374152428033058631510, 7.26459614109627451627208997939, 8.269678588297548132646469576273, 9.222710357598173342635469865952, 10.01039162382509639130361845528, 11.02083640244064833566743878597, 11.87293998489094603838139075387, 13.17331768738502154958108397882

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