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Properties

Label 2-648-9.4-c1-0-1
Degree 22
Conductor 648648
Sign −0.173−0.984i-0.173 - 0.984i
Analytic cond. 5.174305.17430
Root an. cond. 2.274712.27471
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  + (−1 + 1.73i)5-s + (2 + 3.46i)11-s + (1 − 1.73i)13-s − 2·17-s − 4·19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + (3 + 5.19i)29-s + (−4 + 6.92i)31-s + 6·37-s + (−3 + 5.19i)41-s + (−2 − 3.46i)43-s + (3.5 − 6.06i)49-s + 2·53-s − 7.99·55-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (0.603 + 1.04i)11-s + (0.277 − 0.480i)13-s − 0.485·17-s − 0.917·19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + (0.557 + 0.964i)29-s + (−0.718 + 1.24i)31-s + 0.986·37-s + (−0.468 + 0.811i)41-s + (−0.304 − 0.528i)43-s + (0.5 − 0.866i)49-s + 0.274·53-s − 1.07·55-s + ⋯

Functional equation

Λ(s)=(648s/2 ΓC(s) L(s)=((−0.173−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(648s/2 ΓC(s+1/2) L(s)=((−0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 648648    =    23⋅342^{3} \cdot 3^{4}
Sign: −0.173−0.984i-0.173 - 0.984i
Analytic conductor: 5.174305.17430
Root analytic conductor: 2.274712.27471
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ648(433,⋅)\chi_{648} (433, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 648, ( :1/2), −0.173−0.984i)(2,\ 648,\ (\ :1/2),\ -0.173 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 0.722386+0.860906i0.722386 + 0.860906i
L(12)L(\frac12) ≈\approx 0.722386+0.860906i0.722386 + 0.860906i
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+(1−1.73i)T+(−2.5−4.33i)T2 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2}
7 1+(−3.5+6.06i)T2 1 + (-3.5 + 6.06i)T^{2}
11 1+(−2−3.46i)T+(−5.5+9.52i)T2 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1+1.73i)T+(−6.5−11.2i)T2 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+(4−6.92i)T+(−11.5−19.9i)T2 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3−5.19i)T+(−14.5+25.1i)T2 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1+(4−6.92i)T+(−15.5−26.8i)T2 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1+(3−5.19i)T+(−20.5−35.5i)T2 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2}
43 1+(2+3.46i)T+(−21.5+37.2i)T2 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2}
47 1+(−23.5+40.7i)T2 1 + (-23.5 + 40.7i)T^{2}
53 1−2T+53T2 1 - 2T + 53T^{2}
59 1+(−2+3.46i)T+(−29.5−51.0i)T2 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1−1.73i)T+(−30.5+52.8i)T2 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2+3.46i)T+(−33.5−58.0i)T2 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1+(−4−6.92i)T+(−39.5+68.4i)T2 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2}
83 1+(2+3.46i)T+(−41.5+71.8i)T2 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+(1+1.73i)T+(−48.5+84.0i)T2 1 + (1 + 1.73i)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

−10.79136450462894180523473862426, −10.03990724943601265756739972569, −9.091511287807946275349639651779, −8.118841486681787115399695592518, −7.13723303190089553070670520054, −6.58556329768194161716179141707, −5.32243520848985869924111779925, −4.14448695326889604007949906289, −3.22565559483863579379392605700, −1.78851140451292107198070039481, 0.61352758754528296354893035184, 2.32944008006588739182324639947, 3.95848441804194632447127503776, 4.49268408687844431066816523150, 5.94282249146933867362057864076, 6.56281557415033051199783292596, 7.958739800770420305055354198236, 8.572771034830756329105478232988, 9.228785968814071162613509608709, 10.39364945474135883701952701496

Graph of the ZZ-function along the critical line