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Properties

Label 2-45-5.3-c4-0-3
Degree 22
Conductor 4545
Sign 0.525−0.850i0.525 - 0.850i
Analytic cond. 4.651644.65164
Root an. cond. 2.156762.15676
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5 + 5i)2-s − 34i·4-s + 25·5-s + (40 − 40i)7-s + (90 + 90i)8-s + (−125 + 125i)10-s − 100·11-s + (205 + 205i)13-s + 400i·14-s − 356·16-s + (235 − 235i)17-s − 72i·19-s − 850i·20-s + (500 − 500i)22-s + (340 + 340i)23-s + ⋯
L(s)  = 1  + (−1.25 + 1.25i)2-s − 2.12i·4-s + 5-s + (0.816 − 0.816i)7-s + (1.40 + 1.40i)8-s + (−1.25 + 1.25i)10-s − 0.826·11-s + (1.21 + 1.21i)13-s + 2.04i·14-s − 1.39·16-s + (0.813 − 0.813i)17-s − 0.199i·19-s − 2.12i·20-s + (1.03 − 1.03i)22-s + (0.642 + 0.642i)23-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((0.525−0.850i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+2) L(s)=((0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 0.525−0.850i0.525 - 0.850i
Analytic conductor: 4.651644.65164
Root analytic conductor: 2.156762.15676
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ45(28,⋅)\chi_{45} (28, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :2), 0.525−0.850i)(2,\ 45,\ (\ :2),\ 0.525 - 0.850i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.884644+0.493221i0.884644 + 0.493221i
L(12)L(\frac12) ≈\approx 0.884644+0.493221i0.884644 + 0.493221i
L(3)L(3) 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 1−25T 1 - 25T
good2 1+(5−5i)T−16iT2 1 + (5 - 5i)T - 16iT^{2}
7 1+(−40+40i)T−2.40e3iT2 1 + (-40 + 40i)T - 2.40e3iT^{2}
11 1+100T+1.46e4T2 1 + 100T + 1.46e4T^{2}
13 1+(−205−205i)T+2.85e4iT2 1 + (-205 - 205i)T + 2.85e4iT^{2}
17 1+(−235+235i)T−8.35e4iT2 1 + (-235 + 235i)T - 8.35e4iT^{2}
19 1+72iT−1.30e5T2 1 + 72iT - 1.30e5T^{2}
23 1+(−340−340i)T+2.79e5iT2 1 + (-340 - 340i)T + 2.79e5iT^{2}
29 1+450iT−7.07e5T2 1 + 450iT - 7.07e5T^{2}
31 1−428T+9.23e5T2 1 - 428T + 9.23e5T^{2}
37 1+(755−755i)T−1.87e6iT2 1 + (755 - 755i)T - 1.87e6iT^{2}
41 1−950T+2.82e6T2 1 - 950T + 2.82e6T^{2}
43 1+(1.22e3+1.22e3i)T+3.41e6iT2 1 + (1.22e3 + 1.22e3i)T + 3.41e6iT^{2}
47 1+(320−320i)T−4.87e6iT2 1 + (320 - 320i)T - 4.87e6iT^{2}
53 1+(−505−505i)T+7.89e6iT2 1 + (-505 - 505i)T + 7.89e6iT^{2}
59 1+6.30e3iT−1.21e7T2 1 + 6.30e3iT - 1.21e7T^{2}
61 1+3.80e3T+1.38e7T2 1 + 3.80e3T + 1.38e7T^{2}
67 1+(−340+340i)T−2.01e7iT2 1 + (-340 + 340i)T - 2.01e7iT^{2}
71 1+3.40e3T+2.54e7T2 1 + 3.40e3T + 2.54e7T^{2}
73 1+(−415−415i)T+2.83e7iT2 1 + (-415 - 415i)T + 2.83e7iT^{2}
79 1−6.73e3iT−3.89e7T2 1 - 6.73e3iT - 3.89e7T^{2}
83 1+(680+680i)T+4.74e7iT2 1 + (680 + 680i)T + 4.74e7iT^{2}
89 1+2.25e3iT−6.27e7T2 1 + 2.25e3iT - 6.27e7T^{2}
97 1+(−1.61e3+1.61e3i)T−8.85e7iT2 1 + (-1.61e3 + 1.61e3i)T - 8.85e7iT^{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

−15.60902321995802157718247358034, −14.22887761342581863350793123036, −13.60469477727313725774403377547, −11.13863414495670208643152394214, −10.05764281745735347802030116292, −8.978313391807621173776114886713, −7.74570646486791488147425204999, −6.56798480812039893955511375980, −5.18008670430117112402853184422, −1.29531835230208170270457325946, 1.39363567087694261239185012619, 2.88303186380730606771898343364, 5.61584908961168104185922907072, 8.063258574828759974350640315673, 8.842516389332419937877742541108, 10.27816425535131573361315562552, 10.87071480473455952488790464967, 12.32798336888523913000149918694, 13.23243016215544772796885912275, 14.93335667345749484841391268839

Graph of the ZZ-function along the critical line