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Properties

Label 2-12e2-9.2-c2-0-7
Degree 22
Conductor 144144
Sign −0.491+0.870i-0.491 + 0.870i
Analytic cond. 3.923713.92371
Root an. cond. 1.980831.98083
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 + 1.73i)3-s + (−4.5 + 2.59i)5-s + (3.17 − 5.49i)7-s + (2.99 − 8.48i)9-s + (−8.17 − 4.71i)11-s + (−9.84 − 17.0i)13-s + (6.52 − 14.1i)15-s − 1.90i·17-s − 4.69·19-s + (1.74 + 18.9i)21-s + (−8.17 + 4.71i)23-s + (1 − 1.73i)25-s + (7.34 + 25.9i)27-s + (−2.84 − 1.64i)29-s + (−20.5 − 35.5i)31-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + (−0.900 + 0.519i)5-s + (0.453 − 0.785i)7-s + (0.333 − 0.942i)9-s + (−0.743 − 0.429i)11-s + (−0.757 − 1.31i)13-s + (0.434 − 0.943i)15-s − 0.112i·17-s − 0.247·19-s + (0.0832 + 0.903i)21-s + (−0.355 + 0.205i)23-s + (0.0400 − 0.0692i)25-s + (0.272 + 0.962i)27-s + (−0.0982 − 0.0567i)29-s + (−0.662 − 1.14i)31-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((−0.491+0.870i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1) L(s)=((−0.491+0.870i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: −0.491+0.870i-0.491 + 0.870i
Analytic conductor: 3.923713.92371
Root analytic conductor: 1.980831.98083
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ144(65,⋅)\chi_{144} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 144, ( :1), −0.491+0.870i)(2,\ 144,\ (\ :1),\ -0.491 + 0.870i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.172408−0.295463i0.172408 - 0.295463i
L(12)L(\frac12) ≈\approx 0.172408−0.295463i0.172408 - 0.295463i
L(2)L(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+(2.44−1.73i)T 1 + (2.44 - 1.73i)T
good5 1+(4.5−2.59i)T+(12.5−21.6i)T2 1 + (4.5 - 2.59i)T + (12.5 - 21.6i)T^{2}
7 1+(−3.17+5.49i)T+(−24.5−42.4i)T2 1 + (-3.17 + 5.49i)T + (-24.5 - 42.4i)T^{2}
11 1+(8.17+4.71i)T+(60.5+104.i)T2 1 + (8.17 + 4.71i)T + (60.5 + 104. i)T^{2}
13 1+(9.84+17.0i)T+(−84.5+146.i)T2 1 + (9.84 + 17.0i)T + (-84.5 + 146. i)T^{2}
17 1+1.90iT−289T2 1 + 1.90iT - 289T^{2}
19 1+4.69T+361T2 1 + 4.69T + 361T^{2}
23 1+(8.17−4.71i)T+(264.5−458.i)T2 1 + (8.17 - 4.71i)T + (264.5 - 458. i)T^{2}
29 1+(2.84+1.64i)T+(420.5+728.i)T2 1 + (2.84 + 1.64i)T + (420.5 + 728. i)T^{2}
31 1+(20.5+35.5i)T+(−480.5+832.i)T2 1 + (20.5 + 35.5i)T + (-480.5 + 832. i)T^{2}
37 1−17.3T+1.36e3T2 1 - 17.3T + 1.36e3T^{2}
41 1+(53.5−30.9i)T+(840.5−1.45e3i)T2 1 + (53.5 - 30.9i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−0.477+0.826i)T+(−924.5−1.60e3i)T2 1 + (-0.477 + 0.826i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(−12.2−7.05i)T+(1.10e3+1.91e3i)T2 1 + (-12.2 - 7.05i)T + (1.10e3 + 1.91e3i)T^{2}
53 1+9.53iT−2.80e3T2 1 + 9.53iT - 2.80e3T^{2}
59 1+(79.2−45.7i)T+(1.74e3−3.01e3i)T2 1 + (79.2 - 45.7i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−37.5+65.0i)T+(−1.86e3−3.22e3i)T2 1 + (-37.5 + 65.0i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−15.4−26.8i)T+(−2.24e3+3.88e3i)T2 1 + (-15.4 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1+85.9iT−5.04e3T2 1 + 85.9iT - 5.04e3T^{2}
73 1+96.0T+5.32e3T2 1 + 96.0T + 5.32e3T^{2}
79 1+(−14.8+25.7i)T+(−3.12e3−5.40e3i)T2 1 + (-14.8 + 25.7i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−76.1−43.9i)T+(3.44e3+5.96e3i)T2 1 + (-76.1 - 43.9i)T + (3.44e3 + 5.96e3i)T^{2}
89 1+41.3iT−7.92e3T2 1 + 41.3iT - 7.92e3T^{2}
97 1+(47.9−83.0i)T+(−4.70e3−8.14e3i)T2 1 + (47.9 - 83.0i)T + (-4.70e3 - 8.14e3i)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

−12.32493536890250491468456472547, −11.26398465905026015401742623605, −10.70230545087120722465878286628, −9.801104878953477333529258699936, −8.031322823126616782190067622453, −7.28474345165809796533211691652, −5.75716243159949674768872300727, −4.56958643147551926490471684925, −3.32261068894178789500529427228, −0.23865681734576980419040911121, 2.02762009226977346469245053078, 4.45162547544470042864514415365, 5.35216484770036810621889406606, 6.83878757895171972935704247588, 7.83479549471841972144866279190, 8.851959238502752369320578238840, 10.34581530760745546153594307687, 11.56228005899833088163227017937, 12.09208446929514916313846135371, 12.79828574425227898444252765715

Graph of the ZZ-function along the critical line