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Properties

Label 2-405-135.113-c1-0-4
Degree 22
Conductor 405405
Sign 0.379−0.925i0.379 - 0.925i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
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  + (0.0152 + 0.00133i)2-s + (−1.96 − 0.347i)4-s + (1.79 + 1.33i)5-s + (0.211 + 0.148i)7-s + (−0.0589 − 0.0158i)8-s + (0.0255 + 0.0226i)10-s + (−1.49 + 4.11i)11-s + (0.187 + 2.14i)13-s + (0.00301 + 0.00253i)14-s + (3.75 + 1.36i)16-s + (−1.70 + 0.456i)17-s + (4.91 − 2.83i)19-s + (−3.06 − 3.25i)20-s + (−0.0282 + 0.0605i)22-s + (3.32 + 4.74i)23-s + ⋯
L(s)  = 1  + (0.0107 + 0.000940i)2-s + (−0.984 − 0.173i)4-s + (0.802 + 0.596i)5-s + (0.0799 + 0.0559i)7-s + (−0.0208 − 0.00558i)8-s + (0.00806 + 0.00717i)10-s + (−0.451 + 1.23i)11-s + (0.0519 + 0.594i)13-s + (0.000806 + 0.000676i)14-s + (0.939 + 0.341i)16-s + (−0.412 + 0.110i)17-s + (1.12 − 0.651i)19-s + (−0.686 − 0.727i)20-s + (−0.00601 + 0.0129i)22-s + (0.693 + 0.989i)23-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((0.379−0.925i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((0.379−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: 0.379−0.925i0.379 - 0.925i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(368,⋅)\chi_{405} (368, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), 0.379−0.925i)(2,\ 405,\ (\ :1/2),\ 0.379 - 0.925i)

Particular Values

L(1)L(1) ≈\approx 0.970005+0.650613i0.970005 + 0.650613i
L(12)L(\frac12) ≈\approx 0.970005+0.650613i0.970005 + 0.650613i
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)
bad3 1 1
5 1+(−1.79−1.33i)T 1 + (-1.79 - 1.33i)T
good2 1+(−0.0152−0.00133i)T+(1.96+0.347i)T2 1 + (-0.0152 - 0.00133i)T + (1.96 + 0.347i)T^{2}
7 1+(−0.211−0.148i)T+(2.39+6.57i)T2 1 + (-0.211 - 0.148i)T + (2.39 + 6.57i)T^{2}
11 1+(1.49−4.11i)T+(−8.42−7.07i)T2 1 + (1.49 - 4.11i)T + (-8.42 - 7.07i)T^{2}
13 1+(−0.187−2.14i)T+(−12.8+2.25i)T2 1 + (-0.187 - 2.14i)T + (-12.8 + 2.25i)T^{2}
17 1+(1.70−0.456i)T+(14.7−8.5i)T2 1 + (1.70 - 0.456i)T + (14.7 - 8.5i)T^{2}
19 1+(−4.91+2.83i)T+(9.5−16.4i)T2 1 + (-4.91 + 2.83i)T + (9.5 - 16.4i)T^{2}
23 1+(−3.32−4.74i)T+(−7.86+21.6i)T2 1 + (-3.32 - 4.74i)T + (-7.86 + 21.6i)T^{2}
29 1+(3.59−3.01i)T+(5.03−28.5i)T2 1 + (3.59 - 3.01i)T + (5.03 - 28.5i)T^{2}
31 1+(0.912−5.17i)T+(−29.1−10.6i)T2 1 + (0.912 - 5.17i)T + (-29.1 - 10.6i)T^{2}
37 1+(0.837+3.12i)T+(−32.0+18.5i)T2 1 + (0.837 + 3.12i)T + (-32.0 + 18.5i)T^{2}
41 1+(0.241−0.288i)T+(−7.11−40.3i)T2 1 + (0.241 - 0.288i)T + (-7.11 - 40.3i)T^{2}
43 1+(−3.84−8.25i)T+(−27.6+32.9i)T2 1 + (-3.84 - 8.25i)T + (-27.6 + 32.9i)T^{2}
47 1+(−2.29+3.28i)T+(−16.0−44.1i)T2 1 + (-2.29 + 3.28i)T + (-16.0 - 44.1i)T^{2}
53 1+(−8.15+8.15i)T−53iT2 1 + (-8.15 + 8.15i)T - 53iT^{2}
59 1+(10.6−3.86i)T+(45.1−37.9i)T2 1 + (10.6 - 3.86i)T + (45.1 - 37.9i)T^{2}
61 1+(2.10+11.9i)T+(−57.3+20.8i)T2 1 + (2.10 + 11.9i)T + (-57.3 + 20.8i)T^{2}
67 1+(−1.82+0.160i)T+(65.9−11.6i)T2 1 + (-1.82 + 0.160i)T + (65.9 - 11.6i)T^{2}
71 1+(4.44+2.56i)T+(35.5+61.4i)T2 1 + (4.44 + 2.56i)T + (35.5 + 61.4i)T^{2}
73 1+(−3.71+13.8i)T+(−63.2−36.5i)T2 1 + (-3.71 + 13.8i)T + (-63.2 - 36.5i)T^{2}
79 1+(−1.98−2.37i)T+(−13.7+77.7i)T2 1 + (-1.98 - 2.37i)T + (-13.7 + 77.7i)T^{2}
83 1+(−0.432+4.94i)T+(−81.7−14.4i)T2 1 + (-0.432 + 4.94i)T + (-81.7 - 14.4i)T^{2}
89 1+(−1.44−2.50i)T+(−44.5+77.0i)T2 1 + (-1.44 - 2.50i)T + (-44.5 + 77.0i)T^{2}
97 1+(4.27−1.99i)T+(62.3−74.3i)T2 1 + (4.27 - 1.99i)T + (62.3 - 74.3i)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

−11.27807766308572291881528792626, −10.36599060959774239284467987159, −9.475895689691754641730473773565, −9.090746656284520158728110070285, −7.60755925854980515299392007369, −6.78486064291313070112335821027, −5.44977470031089115676618959757, −4.77684288052826768601096985221, −3.32792127684363019341922616540, −1.77346081509742684541433016099, 0.836872879079531810798887576801, 2.87204742447380718936998715550, 4.22199215023551456195996302684, 5.38625881495130920522718092042, 5.91965773192701788223175089374, 7.61582297273030522601915853096, 8.509887703605462983906559084343, 9.165410469360617553172994709547, 10.05900173074457708366173101165, 10.93853824974532208365491633303

Graph of the ZZ-function along the critical line