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Properties

Label 2-55-5.4-c3-0-5
Degree 22
Conductor 5555
Sign 0.597−0.801i0.597 - 0.801i
Analytic cond. 3.245103.24510
Root an. cond. 1.801411.80141
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.245i·2-s + 2.52i·3-s + 7.93·4-s + (−6.68 + 8.96i)5-s + 0.621·6-s + 10.5i·7-s − 3.91i·8-s + 20.6·9-s + (2.20 + 1.64i)10-s + 11·11-s + 20.0i·12-s + 63.0i·13-s + 2.58·14-s + (−22.6 − 16.8i)15-s + 62.5·16-s − 133. i·17-s + ⋯
L(s)  = 1  − 0.0869i·2-s + 0.486i·3-s + 0.992·4-s + (−0.597 + 0.801i)5-s + 0.0422·6-s + 0.567i·7-s − 0.173i·8-s + 0.763·9-s + (0.0697 + 0.0519i)10-s + 0.301·11-s + 0.482i·12-s + 1.34i·13-s + 0.0492·14-s + (−0.389 − 0.290i)15-s + 0.977·16-s − 1.91i·17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.597−0.801i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+3/2) L(s)=((0.597−0.801i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.597−0.801i0.597 - 0.801i
Analytic conductor: 3.245103.24510
Root analytic conductor: 1.801411.80141
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ55(34,⋅)\chi_{55} (34, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :3/2), 0.597−0.801i)(2,\ 55,\ (\ :3/2),\ 0.597 - 0.801i)

Particular Values

L(2)L(2) ≈\approx 1.41583+0.710637i1.41583 + 0.710637i
L(12)L(\frac12) ≈\approx 1.41583+0.710637i1.41583 + 0.710637i
L(52)L(\frac{5}{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)
bad5 1+(6.68−8.96i)T 1 + (6.68 - 8.96i)T
11 1−11T 1 - 11T
good2 1+0.245iT−8T2 1 + 0.245iT - 8T^{2}
3 1−2.52iT−27T2 1 - 2.52iT - 27T^{2}
7 1−10.5iT−343T2 1 - 10.5iT - 343T^{2}
13 1−63.0iT−2.19e3T2 1 - 63.0iT - 2.19e3T^{2}
17 1+133.iT−4.91e3T2 1 + 133. iT - 4.91e3T^{2}
19 1+76.1T+6.85e3T2 1 + 76.1T + 6.85e3T^{2}
23 1+169.iT−1.21e4T2 1 + 169. iT - 1.21e4T^{2}
29 1+202.T+2.43e4T2 1 + 202.T + 2.43e4T^{2}
31 1−191.T+2.97e4T2 1 - 191.T + 2.97e4T^{2}
37 1−21.5iT−5.06e4T2 1 - 21.5iT - 5.06e4T^{2}
41 1−305.T+6.89e4T2 1 - 305.T + 6.89e4T^{2}
43 1+285.iT−7.95e4T2 1 + 285. iT - 7.95e4T^{2}
47 1−123.iT−1.03e5T2 1 - 123. iT - 1.03e5T^{2}
53 1−480.iT−1.48e5T2 1 - 480. iT - 1.48e5T^{2}
59 1+364.T+2.05e5T2 1 + 364.T + 2.05e5T^{2}
61 1−9.11T+2.26e5T2 1 - 9.11T + 2.26e5T^{2}
67 1+568.iT−3.00e5T2 1 + 568. iT - 3.00e5T^{2}
71 1+157.T+3.57e5T2 1 + 157.T + 3.57e5T^{2}
73 1−212.iT−3.89e5T2 1 - 212. iT - 3.89e5T^{2}
79 1+792.T+4.93e5T2 1 + 792.T + 4.93e5T^{2}
83 1+587.iT−5.71e5T2 1 + 587. iT - 5.71e5T^{2}
89 1+698.T+7.04e5T2 1 + 698.T + 7.04e5T^{2}
97 1−1.83e3iT−9.12e5T2 1 - 1.83e3iT - 9.12e5T^{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

−15.17170670624021418912404253981, −14.18280251066818275716459794749, −12.31136285392691955326083029534, −11.48104689896880138243527215911, −10.52539903083314285894660361750, −9.200085189964228357109866776729, −7.36455294690950415527808886130, −6.47966976713271647140241072471, −4.29534180238656807250389115103, −2.51496888048124476394422691739, 1.41068921700332330229546635928, 3.89167045678883275687492191662, 5.93538330656280125107810786661, 7.39717005479031148221783000671, 8.187263697860861615547186794245, 10.14329599760203456992339344367, 11.25759200325413225447026582266, 12.57937617922278048576404192599, 13.08025313502288710352283941419, 15.01514863537872588164457238477

Graph of the ZZ-function along the critical line