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Properties

Label 2-12e2-144.11-c1-0-9
Degree $2$
Conductor $144$
Sign $0.335 + 0.942i$
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  + (−1.39 + 0.225i)2-s + (−1.36 + 1.05i)3-s + (1.89 − 0.628i)4-s + (−2.78 − 0.746i)5-s + (1.67 − 1.78i)6-s + (1.16 − 2.02i)7-s + (−2.50 + 1.30i)8-s + (0.753 − 2.90i)9-s + (4.05 + 0.415i)10-s + (5.53 − 1.48i)11-s + (−1.93 + 2.87i)12-s + (−3.90 − 1.04i)13-s + (−1.17 + 3.08i)14-s + (4.60 − 1.93i)15-s + (3.20 − 2.38i)16-s − 6.45i·17-s + ⋯
L(s)  = 1  + (−0.987 + 0.159i)2-s + (−0.790 + 0.611i)3-s + (0.949 − 0.314i)4-s + (−1.24 − 0.333i)5-s + (0.683 − 0.730i)6-s + (0.440 − 0.763i)7-s + (−0.887 + 0.461i)8-s + (0.251 − 0.967i)9-s + (1.28 + 0.131i)10-s + (1.67 − 0.447i)11-s + (−0.558 + 0.829i)12-s + (−1.08 − 0.290i)13-s + (−0.313 + 0.824i)14-s + (1.19 − 0.498i)15-s + (0.802 − 0.596i)16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.335 + 0.942i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.335 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323110 - 0.227916i\)
\(L(\frac12)\) \(\approx\) \(0.323110 - 0.227916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.225i)T \)
3 \( 1 + (1.36 - 1.05i)T \)
good5 \( 1 + (2.78 + 0.746i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.16 + 2.02i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.53 + 1.48i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.90 + 1.04i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 6.45iT - 17T^{2} \)
19 \( 1 + (1.50 + 1.50i)T + 19iT^{2} \)
23 \( 1 + (0.0418 - 0.0241i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.08 - 1.36i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (1.65 - 0.952i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.489 - 0.489i)T + 37iT^{2} \)
41 \( 1 + (-0.0155 - 0.0269i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.01 + 3.80i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.0913 + 0.158i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.62 + 6.62i)T - 53iT^{2} \)
59 \( 1 + (-1.11 + 4.15i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.71 + 6.39i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.216 + 0.808i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.04iT - 71T^{2} \)
73 \( 1 - 4.74iT - 73T^{2} \)
79 \( 1 + (-7.29 - 4.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.20 - 11.9i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.85T + 89T^{2} \)
97 \( 1 + (-3.29 + 5.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.25400500492880877235881242893, −11.58165520080672154269828111604, −11.03359427719893126090852289885, −9.747838399369550712318145221957, −8.892535742127168504410794089005, −7.53187640281573920414744191821, −6.76285174057287760833573620927, −5.04707143667454557275566780735, −3.76751611932508054265942669089, −0.60872501127766903055681199334, 1.86300766391380955148194981557, 4.07602121879338611532859137857, 6.06162321454245023108288229394, 7.11546558744579052614661711187, 7.931022686598030205387250419296, 9.042537297281798947547978581241, 10.43050443944445927597160885766, 11.53567641687946449563194226079, 11.91140523949891595102773207439, 12.62013631044794417291694662464

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