Properties

Label 2-12e2-144.85-c1-0-17
Degree $2$
Conductor $144$
Sign $-0.274 + 0.961i$
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  + (−0.0882 − 1.41i)2-s + (1.08 − 1.34i)3-s + (−1.98 + 0.249i)4-s + (2.90 − 0.777i)5-s + (−1.99 − 1.41i)6-s + (−1.04 + 0.603i)7-s + (0.526 + 2.77i)8-s + (−0.636 − 2.93i)9-s + (−1.35 − 4.02i)10-s + (−1.36 + 5.09i)11-s + (−1.82 + 2.94i)12-s + (−0.541 − 2.02i)13-s + (0.944 + 1.42i)14-s + (2.10 − 4.75i)15-s + (3.87 − 0.989i)16-s − 3.20·17-s + ⋯
L(s)  = 1  + (−0.0624 − 0.998i)2-s + (0.627 − 0.778i)3-s + (−0.992 + 0.124i)4-s + (1.29 − 0.347i)5-s + (−0.816 − 0.577i)6-s + (−0.395 + 0.228i)7-s + (0.186 + 0.982i)8-s + (−0.212 − 0.977i)9-s + (−0.427 − 1.27i)10-s + (−0.411 + 1.53i)11-s + (−0.525 + 0.850i)12-s + (−0.150 − 0.560i)13-s + (0.252 + 0.380i)14-s + (0.543 − 1.22i)15-s + (0.968 − 0.247i)16-s − 0.777·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.274 + 0.961i$
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: \((2,\ 144,\ (\ :1/2),\ -0.274 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783065 - 1.03780i\)
\(L(\frac12)\) \(\approx\) \(0.783065 - 1.03780i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0882 + 1.41i)T \)
3 \( 1 + (-1.08 + 1.34i)T \)
good5 \( 1 + (-2.90 + 0.777i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.04 - 0.603i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.36 - 5.09i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.541 + 2.02i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 + (1.87 - 1.87i)T - 19iT^{2} \)
23 \( 1 + (-3.61 - 2.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.94 - 2.12i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-1.39 + 2.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.10 + 5.10i)T + 37iT^{2} \)
41 \( 1 + (-9.93 - 5.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.293 - 1.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.84 + 3.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.613 + 0.613i)T + 53iT^{2} \)
59 \( 1 + (11.8 - 3.16i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.40 + 1.98i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.64 + 9.86i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 - 2.87iT - 73T^{2} \)
79 \( 1 + (0.913 + 1.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.82 + 0.757i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.11iT - 89T^{2} \)
97 \( 1 + (3.06 + 5.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.77559602946427842484872924727, −12.26492177968360867282754617359, −10.59024188597534893881701216460, −9.618264737636571944779850196375, −9.030129682454791134713170834493, −7.70492745907375292904589609804, −6.20811921306406035102858313046, −4.76655259702456139243019582725, −2.79304099289609435136613768132, −1.75780351691685219990523202080, 2.91901473710280042842707345700, 4.59050127597776537987457673483, 5.85238987410086229361729069183, 6.79829279168163552342176957525, 8.432041805543156531069364556406, 9.092720993531156162672435201721, 10.10004874177480008245092712434, 10.87248551626248372457747816270, 13.04487124799699816028226245687, 13.87308281894715173543835633798

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