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Properties

Label 2-12e3-9.4-c1-0-11
Degree 22
Conductor 17281728
Sign 0.996−0.0825i0.996 - 0.0825i
Analytic cond. 13.798113.7981
Root an. cond. 3.714583.71458
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)5-s + (1.72 + 2.98i)7-s + (0.724 + 1.25i)11-s + (2.94 − 5.10i)13-s − 4.89·17-s + 4·19-s + (2.72 − 4.71i)23-s + (2 + 3.46i)25-s + (−0.0505 − 0.0874i)29-s + (1.27 − 2.20i)31-s + 3.44·35-s + 0.898·37-s + (−5.94 + 10.3i)41-s + (1.17 + 2.03i)43-s + (3.17 + 5.49i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.651 + 1.12i)7-s + (0.218 + 0.378i)11-s + (0.818 − 1.41i)13-s − 1.18·17-s + 0.917·19-s + (0.568 − 0.984i)23-s + (0.400 + 0.692i)25-s + (−0.00937 − 0.0162i)29-s + (0.229 − 0.396i)31-s + 0.583·35-s + 0.147·37-s + (−0.929 + 1.60i)41-s + (0.179 + 0.310i)43-s + (0.463 + 0.801i)47-s + ⋯

Functional equation

Λ(s)=(1728s/2 ΓC(s) L(s)=((0.996−0.0825i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0825i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1728s/2 ΓC(s+1/2) L(s)=((0.996−0.0825i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 0.996−0.0825i0.996 - 0.0825i
Analytic conductor: 13.798113.7981
Root analytic conductor: 3.714583.71458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1728(577,⋅)\chi_{1728} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1728, ( :1/2), 0.996−0.0825i)(2,\ 1728,\ (\ :1/2),\ 0.996 - 0.0825i)

Particular Values

L(1)L(1) ≈\approx 2.0659981702.065998170
L(12)L(\frac12) ≈\approx 2.0659981702.065998170
L(32)L(\frac{3}{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+(−0.5+0.866i)T+(−2.5−4.33i)T2 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2}
7 1+(−1.72−2.98i)T+(−3.5+6.06i)T2 1 + (-1.72 - 2.98i)T + (-3.5 + 6.06i)T^{2}
11 1+(−0.724−1.25i)T+(−5.5+9.52i)T2 1 + (-0.724 - 1.25i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2.94+5.10i)T+(−6.5−11.2i)T2 1 + (-2.94 + 5.10i)T + (-6.5 - 11.2i)T^{2}
17 1+4.89T+17T2 1 + 4.89T + 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+(−2.72+4.71i)T+(−11.5−19.9i)T2 1 + (-2.72 + 4.71i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.0505+0.0874i)T+(−14.5+25.1i)T2 1 + (0.0505 + 0.0874i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.27+2.20i)T+(−15.5−26.8i)T2 1 + (-1.27 + 2.20i)T + (-15.5 - 26.8i)T^{2}
37 1−0.898T+37T2 1 - 0.898T + 37T^{2}
41 1+(5.94−10.3i)T+(−20.5−35.5i)T2 1 + (5.94 - 10.3i)T + (-20.5 - 35.5i)T^{2}
43 1+(−1.17−2.03i)T+(−21.5+37.2i)T2 1 + (-1.17 - 2.03i)T + (-21.5 + 37.2i)T^{2}
47 1+(−3.17−5.49i)T+(−23.5+40.7i)T2 1 + (-3.17 - 5.49i)T + (-23.5 + 40.7i)T^{2}
53 1−8.89T+53T2 1 - 8.89T + 53T^{2}
59 1+(−7.17+12.4i)T+(−29.5−51.0i)T2 1 + (-7.17 + 12.4i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.94+6.84i)T+(−30.5+52.8i)T2 1 + (3.94 + 6.84i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.17−10.6i)T+(−33.5−58.0i)T2 1 + (6.17 - 10.6i)T + (-33.5 - 58.0i)T^{2}
71 1−7.79T+71T2 1 - 7.79T + 71T^{2}
73 1+4.89T+73T2 1 + 4.89T + 73T^{2}
79 1+(−6.72−11.6i)T+(−39.5+68.4i)T2 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.275+0.476i)T+(−41.5+71.8i)T2 1 + (0.275 + 0.476i)T + (-41.5 + 71.8i)T^{2}
89 1−12.8T+89T2 1 - 12.8T + 89T^{2}
97 1+(−1.94−3.37i)T+(−48.5+84.0i)T2 1 + (-1.94 - 3.37i)T + (-48.5 + 84.0i)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

−9.192549985877580440630255961023, −8.535526474325837693306333513048, −8.014348212281432522867110797302, −6.88701627700647538406160476821, −5.97361148878903154157960562778, −5.24634318619143262840983201375, −4.58423533164180212757894368288, −3.23574429797695283510368643474, −2.29480020105080556231685410458, −1.07336079612621537608805950955, 1.04951655921322591573620054157, 2.13356210915103850722693014870, 3.55066214829680230130596966838, 4.21386504088834079915415819672, 5.14450345167735074413660997233, 6.27139523363594037707538634087, 6.98654862241536531562171246045, 7.51363841545800908509403922792, 8.785171700080501159457487586262, 9.043556190990965701859040056762

Graph of the ZZ-function along the critical line