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Properties

Label 2-12e3-9.2-c2-0-6
Degree 22
Conductor 17281728
Sign −0.996−0.0825i-0.996 - 0.0825i
Analytic cond. 47.084547.0845
Root an. cond. 6.861826.86182
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  + (−4.5 + 2.59i)5-s + (−3.17 + 5.49i)7-s + (8.17 + 4.71i)11-s + (9.84 + 17.0i)13-s + 1.90i·17-s − 4.69·19-s + (−8.17 + 4.71i)23-s + (1 − 1.73i)25-s + (−2.84 − 1.64i)29-s + (20.5 + 35.5i)31-s − 32.9i·35-s − 17.3·37-s + (53.5 − 30.9i)41-s + (0.477 − 0.826i)43-s + (12.2 + 7.05i)47-s + ⋯
L(s)  = 1  + (−0.900 + 0.519i)5-s + (−0.453 + 0.785i)7-s + (0.743 + 0.429i)11-s + (0.757 + 1.31i)13-s + 0.112i·17-s − 0.247·19-s + (−0.355 + 0.205i)23-s + (0.0400 − 0.0692i)25-s + (−0.0982 − 0.0567i)29-s + (0.662 + 1.14i)31-s − 0.942i·35-s − 0.467·37-s + (1.30 − 0.754i)41-s + (0.0110 − 0.0192i)43-s + (0.259 + 0.150i)47-s + ⋯

Functional equation

Λ(s)=(1728s/2 ΓC(s) L(s)=((−0.996−0.0825i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0825i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(1728s/2 ΓC(s+1) L(s)=((−0.996−0.0825i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0825i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17281728    =    26⋅332^{6} \cdot 3^{3}
Sign: −0.996−0.0825i-0.996 - 0.0825i
Analytic conductor: 47.084547.0845
Root analytic conductor: 6.861826.86182
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ1728(1601,⋅)\chi_{1728} (1601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1728, ( :1), −0.996−0.0825i)(2,\ 1728,\ (\ :1),\ -0.996 - 0.0825i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.0020644231.002064423
L(12)L(\frac12) ≈\approx 1.0020644231.002064423
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 1
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

−9.259194048593712687149113751549, −8.899520608950360891105715732607, −7.947246411682917124997718494866, −6.98118695231615993984429084785, −6.51154284064419864179825496595, −5.58327428287216713753709257361, −4.28168045833499819299540584045, −3.77041957431160221570469650991, −2.67677052810280284563197267216, −1.49580066109554924616755529604, 0.31374660764053104110659780511, 1.09271089670763644727626470940, 2.85423161508242778806935909218, 3.87407344999795321826835196252, 4.26462106656825015524961676917, 5.57259696143950180652037805993, 6.32682482926181371060231435979, 7.26654241204159889011395763101, 8.085339873370891692572286424889, 8.531221518532827374158424288434

Graph of the ZZ-function along the critical line