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Properties

Label 2-40-40.37-c2-0-7
Degree 22
Conductor 4040
Sign −0.174+0.984i-0.174 + 0.984i
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  + (0.376 − 1.96i)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 + (−4.30 + 6.74i)8-s − 7.08i·9-s + (−9.39 − 3.43i)10-s + 4.01i·11-s + (2.18 + 5.07i)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  + (0.188 − 0.982i)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 + (−0.538 + 0.842i)8-s − 0.787i·9-s + (−0.939 − 0.343i)10-s + 0.365i·11-s + (0.182 + 0.423i)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)=((−0.174+0.984i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(40s/2 ΓC(s+1) L(s)=((−0.174+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4040    =    23⋅52^{3} \cdot 5
Sign: −0.174+0.984i-0.174 + 0.984i
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), −0.174+0.984i)(2,\ 40,\ (\ :1),\ -0.174 + 0.984i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.685346−0.817632i0.685346 - 0.817632i
L(12)L(\frac12) ≈\approx 0.685346−0.817632i0.685346 - 0.817632i
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+(−0.376+1.96i)T 1 + (-0.376 + 1.96i)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

−15.41464707505327939104578391050, −14.24682247890604941995045022866, −12.83738684804644557253367181097, −12.03265746395214291190423942213, −11.21247707349386498489225303926, −9.265731312929913675324981149489, −8.534081435595439353902316706645, −5.86050524401683726809087358587, −4.50213454319225510339654013382, −1.71340585576837200706113832598, 4.08477870989734593969642599673, 5.66624847404177367050775249227, 7.25189160141571558037832664134, 8.290880986586959322262782367733, 10.42689097676422202929687231271, 11.06389980171498153397616751204, 13.28504075093105623459820413545, 14.10693900150405982363530706277, 15.05256794949393564274397061208, 16.26136033886806312542870582570

Graph of the ZZ-function along the critical line