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Properties

Label 2-40-40.37-c2-0-1
Degree 22
Conductor 4040
Sign −0.0242−0.999i-0.0242 - 0.999i
Analytic cond. 1.089921.08992
Root an. cond. 1.043991.04399
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  + (−0.665 + 1.88i)2-s + (3.60 + 3.60i)3-s + (−3.11 − 2.51i)4-s + (−2.34 − 4.41i)5-s + (−9.20 + 4.40i)6-s + (1.47 + 1.47i)7-s + (6.80 − 4.20i)8-s + 17.0i·9-s + (9.88 − 1.48i)10-s − 11.3i·11-s + (−2.18 − 20.2i)12-s + (3.17 + 3.17i)13-s + (−3.77 + 1.80i)14-s + (7.46 − 24.3i)15-s + (3.39 + 15.6i)16-s + (−9.94 − 9.94i)17-s + ⋯
L(s)  = 1  + (−0.332 + 0.943i)2-s + (1.20 + 1.20i)3-s + (−0.778 − 0.627i)4-s + (−0.469 − 0.883i)5-s + (−1.53 + 0.733i)6-s + (0.211 + 0.211i)7-s + (0.850 − 0.525i)8-s + 1.89i·9-s + (0.988 − 0.148i)10-s − 1.02i·11-s + (−0.181 − 1.69i)12-s + (0.244 + 0.244i)13-s + (−0.269 + 0.128i)14-s + (0.497 − 1.62i)15-s + (0.212 + 0.977i)16-s + (−0.585 − 0.585i)17-s + ⋯

Functional equation

Λ(s)=(40s/2 ΓC(s) L(s)=((−0.0242−0.999i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0242 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(40s/2 ΓC(s+1) L(s)=((−0.0242−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0242 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4040    =    23⋅52^{3} \cdot 5
Sign: −0.0242−0.999i-0.0242 - 0.999i
Analytic conductor: 1.089921.08992
Root analytic conductor: 1.043991.04399
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ40(37,⋅)\chi_{40} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 40, ( :1), −0.0242−0.999i)(2,\ 40,\ (\ :1),\ -0.0242 - 0.999i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.786780+0.806131i0.786780 + 0.806131i
L(12)L(\frac12) ≈\approx 0.786780+0.806131i0.786780 + 0.806131i
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+(0.665−1.88i)T 1 + (0.665 - 1.88i)T
5 1+(2.34+4.41i)T 1 + (2.34 + 4.41i)T
good3 1+(−3.60−3.60i)T+9iT2 1 + (-3.60 - 3.60i)T + 9iT^{2}
7 1+(−1.47−1.47i)T+49iT2 1 + (-1.47 - 1.47i)T + 49iT^{2}
11 1+11.3iT−121T2 1 + 11.3iT - 121T^{2}
13 1+(−3.17−3.17i)T+169iT2 1 + (-3.17 - 3.17i)T + 169iT^{2}
17 1+(9.94+9.94i)T+289iT2 1 + (9.94 + 9.94i)T + 289iT^{2}
19 1−11.2T+361T2 1 - 11.2T + 361T^{2}
23 1+(−1.67+1.67i)T−529iT2 1 + (-1.67 + 1.67i)T - 529iT^{2}
29 1+41.1T+841T2 1 + 41.1T + 841T^{2}
31 1+29.2T+961T2 1 + 29.2T + 961T^{2}
37 1+(−8.60+8.60i)T−1.36e3iT2 1 + (-8.60 + 8.60i)T - 1.36e3iT^{2}
41 1+19.6T+1.68e3T2 1 + 19.6T + 1.68e3T^{2}
43 1+(−25.1−25.1i)T+1.84e3iT2 1 + (-25.1 - 25.1i)T + 1.84e3iT^{2}
47 1+(−41.7−41.7i)T+2.20e3iT2 1 + (-41.7 - 41.7i)T + 2.20e3iT^{2}
53 1+(2.16+2.16i)T+2.80e3iT2 1 + (2.16 + 2.16i)T + 2.80e3iT^{2}
59 1+38.9T+3.48e3T2 1 + 38.9T + 3.48e3T^{2}
61 1−87.5iT−3.72e3T2 1 - 87.5iT - 3.72e3T^{2}
67 1+(−31.1+31.1i)T−4.48e3iT2 1 + (-31.1 + 31.1i)T - 4.48e3iT^{2}
71 1−134.T+5.04e3T2 1 - 134.T + 5.04e3T^{2}
73 1+(26.1−26.1i)T−5.32e3iT2 1 + (26.1 - 26.1i)T - 5.32e3iT^{2}
79 1−23.1iT−6.24e3T2 1 - 23.1iT - 6.24e3T^{2}
83 1+(−68.3−68.3i)T+6.88e3iT2 1 + (-68.3 - 68.3i)T + 6.88e3iT^{2}
89 1+75.2iT−7.92e3T2 1 + 75.2iT - 7.92e3T^{2}
97 1+(−57.5−57.5i)T+9.40e3iT2 1 + (-57.5 - 57.5i)T + 9.40e3iT^{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

−16.07777353959742085805095212288, −15.32592692504584982730084812473, −14.24741543844654770597697283314, −13.29553320211968589411967854502, −11.06176399351555750560206940570, −9.389167021615519454150414896763, −8.834995416467595895501065831178, −7.77381618285633683633626704079, −5.29856854435201720432563050925, −3.92726453302330650844516521120, 2.07995624338543992283468207048, 3.64595090523606552170010035148, 7.15236491757152050546986656613, 7.966376461843252639970224636735, 9.334764786167300139979279527993, 10.86454961618627660147231789022, 12.20440563218956398657937915630, 13.17982290420087956360123876789, 14.20594779635437694514919023627, 15.18962655491925691943892456207

Graph of the ZZ-function along the critical line