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Properties

Label 2-40-40.37-c2-0-0
Degree 22
Conductor 4040
Sign 0.572−0.819i0.572 - 0.819i
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  + (−1.96 + 0.376i)2-s + (0.977 + 0.977i)3-s + (3.71 − 1.47i)4-s + (−0.801 + 4.93i)5-s + (−2.28 − 1.55i)6-s + (8.39 + 8.39i)7-s + (−6.74 + 4.30i)8-s − 7.08i·9-s + (−0.284 − 9.99i)10-s − 4.01i·11-s + (5.07 + 2.18i)12-s + (−11.0 − 11.0i)13-s + (−19.6 − 13.3i)14-s + (−5.60 + 4.04i)15-s + (11.6 − 10.9i)16-s + (−3.79 − 3.79i)17-s + ⋯
L(s)  = 1  + (−0.982 + 0.188i)2-s + (0.325 + 0.325i)3-s + (0.929 − 0.369i)4-s + (−0.160 + 0.987i)5-s + (−0.381 − 0.258i)6-s + (1.19 + 1.19i)7-s + (−0.842 + 0.538i)8-s − 0.787i·9-s + (−0.0284 − 0.999i)10-s − 0.365i·11-s + (0.423 + 0.182i)12-s + (−0.852 − 0.852i)13-s + (−1.40 − 0.952i)14-s + (−0.373 + 0.269i)15-s + (0.726 − 0.687i)16-s + (−0.223 − 0.223i)17-s + ⋯

Functional equation

Λ(s)=(40s/2 ΓC(s) L(s)=((0.572−0.819i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(40s/2 ΓC(s+1) L(s)=((0.572−0.819i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4040    =    23⋅52^{3} \cdot 5
Sign: 0.572−0.819i0.572 - 0.819i
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), 0.572−0.819i)(2,\ 40,\ (\ :1),\ 0.572 - 0.819i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.728087+0.379440i0.728087 + 0.379440i
L(12)L(\frac12) ≈\approx 0.728087+0.379440i0.728087 + 0.379440i
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+(1.96−0.376i)T 1 + (1.96 - 0.376i)T
5 1+(0.801−4.93i)T 1 + (0.801 - 4.93i)T
good3 1+(−0.977−0.977i)T+9iT2 1 + (-0.977 - 0.977i)T + 9iT^{2}
7 1+(−8.39−8.39i)T+49iT2 1 + (-8.39 - 8.39i)T + 49iT^{2}
11 1+4.01iT−121T2 1 + 4.01iT - 121T^{2}
13 1+(11.0+11.0i)T+169iT2 1 + (11.0 + 11.0i)T + 169iT^{2}
17 1+(3.79+3.79i)T+289iT2 1 + (3.79 + 3.79i)T + 289iT^{2}
19 1−15.9T+361T2 1 - 15.9T + 361T^{2}
23 1+(−1.86+1.86i)T−529iT2 1 + (-1.86 + 1.86i)T - 529iT^{2}
29 1+0.468T+841T2 1 + 0.468T + 841T^{2}
31 1+17.3T+961T2 1 + 17.3T + 961T^{2}
37 1+(−22.1+22.1i)T−1.36e3iT2 1 + (-22.1 + 22.1i)T - 1.36e3iT^{2}
41 1−37.0T+1.68e3T2 1 - 37.0T + 1.68e3T^{2}
43 1+(17.1+17.1i)T+1.84e3iT2 1 + (17.1 + 17.1i)T + 1.84e3iT^{2}
47 1+(6.31+6.31i)T+2.20e3iT2 1 + (6.31 + 6.31i)T + 2.20e3iT^{2}
53 1+(39.8+39.8i)T+2.80e3iT2 1 + (39.8 + 39.8i)T + 2.80e3iT^{2}
59 1−50.6T+3.48e3T2 1 - 50.6T + 3.48e3T^{2}
61 1−73.6iT−3.72e3T2 1 - 73.6iT - 3.72e3T^{2}
67 1+(77.6−77.6i)T−4.48e3iT2 1 + (77.6 - 77.6i)T - 4.48e3iT^{2}
71 1+78.3T+5.04e3T2 1 + 78.3T + 5.04e3T^{2}
73 1+(46.0−46.0i)T−5.32e3iT2 1 + (46.0 - 46.0i)T - 5.32e3iT^{2}
79 1−31.6iT−6.24e3T2 1 - 31.6iT - 6.24e3T^{2}
83 1+(−84.9−84.9i)T+6.88e3iT2 1 + (-84.9 - 84.9i)T + 6.88e3iT^{2}
89 1+92.8iT−7.92e3T2 1 + 92.8iT - 7.92e3T^{2}
97 1+(85.3+85.3i)T+9.40e3iT2 1 + (85.3 + 85.3i)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

−15.92487894084579303689501918356, −14.93200620850232596676388902214, −14.57719169541953968553787245566, −12.03908203402306474565663879204, −11.16602204789449274440304601155, −9.815282873834098907717211867657, −8.621694223771485524766838757256, −7.41480924646251503671946136048, −5.69822813448357183650096673658, −2.73203339004127467286042670167, 1.61840469457882146303169509298, 4.64895682750197650604453595481, 7.36263101960122504856868056212, 8.031401428184193773325856747379, 9.428582085973932025583416684951, 10.84660326870164440151512645452, 11.94642182721100631186389956628, 13.35254595850387471962253621893, 14.59378114489539997826701686204, 16.26125780390559840544619392194

Graph of the ZZ-function along the critical line