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Properties

Label 2-12e2-9.4-c3-0-11
Degree 22
Conductor 144144
Sign 0.173+0.984i0.173 + 0.984i
Analytic cond. 8.496278.49627
Root an. cond. 2.914832.91483
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.19i·3-s + (4.5 − 7.79i)5-s + (−15.5 − 26.8i)7-s − 27·9-s + (−7.5 − 12.9i)11-s + (18.5 − 32.0i)13-s + (40.5 + 23.3i)15-s − 42·17-s + 28·19-s + (139.5 − 80.5i)21-s + (97.5 − 168. i)23-s + (22 + 38.1i)25-s − 140. i·27-s + (−55.5 − 96.1i)29-s + (−102.5 + 177. i)31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.402 − 0.697i)5-s + (−0.836 − 1.44i)7-s − 9-s + (−0.205 − 0.356i)11-s + (0.394 − 0.683i)13-s + (0.697 + 0.402i)15-s − 0.599·17-s + 0.338·19-s + (1.44 − 0.836i)21-s + (0.883 − 1.53i)23-s + (0.175 + 0.304i)25-s − 1.00i·27-s + (−0.355 − 0.615i)29-s + (−0.593 + 1.02i)31-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+3/2) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: 0.173+0.984i0.173 + 0.984i
Analytic conductor: 8.496278.49627
Root analytic conductor: 2.914832.91483
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ144(49,⋅)\chi_{144} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 144, ( :3/2), 0.173+0.984i)(2,\ 144,\ (\ :3/2),\ 0.173 + 0.984i)

Particular Values

L(2)L(2) ≈\approx 0.896013−0.751844i0.896013 - 0.751844i
L(12)L(\frac12) ≈\approx 0.896013−0.751844i0.896013 - 0.751844i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1−5.19iT 1 - 5.19iT
good5 1+(−4.5+7.79i)T+(−62.5−108.i)T2 1 + (-4.5 + 7.79i)T + (-62.5 - 108. i)T^{2}
7 1+(15.5+26.8i)T+(−171.5+297.i)T2 1 + (15.5 + 26.8i)T + (-171.5 + 297. i)T^{2}
11 1+(7.5+12.9i)T+(−665.5+1.15e3i)T2 1 + (7.5 + 12.9i)T + (-665.5 + 1.15e3i)T^{2}
13 1+(−18.5+32.0i)T+(−1.09e3−1.90e3i)T2 1 + (-18.5 + 32.0i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1+42T+4.91e3T2 1 + 42T + 4.91e3T^{2}
19 1−28T+6.85e3T2 1 - 28T + 6.85e3T^{2}
23 1+(−97.5+168.i)T+(−6.08e3−1.05e4i)T2 1 + (-97.5 + 168. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(55.5+96.1i)T+(−1.21e4+2.11e4i)T2 1 + (55.5 + 96.1i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+(102.5−177.i)T+(−1.48e4−2.57e4i)T2 1 + (102.5 - 177. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+166T+5.06e4T2 1 + 166T + 5.06e4T^{2}
41 1+(−130.5+226.i)T+(−3.44e4−5.96e4i)T2 1 + (-130.5 + 226. i)T + (-3.44e4 - 5.96e4i)T^{2}
43 1+(21.5+37.2i)T+(−3.97e4+6.88e4i)T2 1 + (21.5 + 37.2i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−88.5−153.i)T+(−5.19e4+8.99e4i)T2 1 + (-88.5 - 153. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1−114T+1.48e5T2 1 - 114T + 1.48e5T^{2}
59 1+(−79.5+137.i)T+(−1.02e5−1.77e5i)T2 1 + (-79.5 + 137. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(95.5+165.i)T+(−1.13e5+1.96e5i)T2 1 + (95.5 + 165. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(210.5−364.i)T+(−1.50e5−2.60e5i)T2 1 + (210.5 - 364. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+156T+3.57e5T2 1 + 156T + 3.57e5T^{2}
73 1−182T+3.89e5T2 1 - 182T + 3.89e5T^{2}
79 1+(−566.5−981.i)T+(−2.46e5+4.26e5i)T2 1 + (-566.5 - 981. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+(541.5+937.i)T+(−2.85e5+4.95e5i)T2 1 + (541.5 + 937. i)T + (-2.85e5 + 4.95e5i)T^{2}
89 1+1.05e3T+7.04e5T2 1 + 1.05e3T + 7.04e5T^{2}
97 1+(−450.5−780.i)T+(−4.56e5+7.90e5i)T2 1 + (-450.5 - 780. i)T + (-4.56e5 + 7.90e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−12.63454411533842024513409871731, −10.93515162410398207770947964488, −10.44434004476588536725363606051, −9.387684353934934106698772709325, −8.483692024779758458543731784478, −6.93840614120262923114859104892, −5.57359751176674089642378312113, −4.37782043660692309743559654858, −3.20984528326290633803218854987, −0.56050723159954526261225446280, 1.99972156492406522762971089083, 3.09811646190709306470295566380, 5.51497934120753494774056330432, 6.39345413794908118157227988492, 7.29865575009638333516137325121, 8.792069137818794532315345192120, 9.535408448848577925756927779509, 11.08685161137955572248072520465, 11.93359629859719486312321824100, 12.90701471902972991806865397023

Graph of the ZZ-function along the critical line