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Properties

Label 2-275-5.4-c3-0-6
Degree 22
Conductor 275275
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 16.225516.2255
Root an. cond. 4.028094.02809
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  + 2.73i·2-s + 7.92i·3-s + 0.535·4-s − 21.6·6-s + 3.07i·7-s + 23.3i·8-s − 35.8·9-s − 11·11-s + 4.24i·12-s − 5.35i·13-s − 8.39·14-s − 59.4·16-s − 41.2i·17-s − 97.9i·18-s − 139.·19-s + ⋯
L(s)  = 1  + 0.965i·2-s + 1.52i·3-s + 0.0669·4-s − 1.47·6-s + 0.165i·7-s + 1.03i·8-s − 1.32·9-s − 0.301·11-s + 0.102i·12-s − 0.114i·13-s − 0.160·14-s − 0.928·16-s − 0.588i·17-s − 1.28i·18-s − 1.68·19-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+3/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 16.225516.2255
Root analytic conductor: 4.028094.02809
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ275(199,⋅)\chi_{275} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 275, ( :3/2), −0.447+0.894i)(2,\ 275,\ (\ :3/2),\ -0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.4116928351.411692835
L(12)L(\frac12) ≈\approx 1.4116928351.411692835
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 1
11 1+11T 1 + 11T
good2 1−2.73iT−8T2 1 - 2.73iT - 8T^{2}
3 1−7.92iT−27T2 1 - 7.92iT - 27T^{2}
7 1−3.07iT−343T2 1 - 3.07iT - 343T^{2}
13 1+5.35iT−2.19e3T2 1 + 5.35iT - 2.19e3T^{2}
17 1+41.2iT−4.91e3T2 1 + 41.2iT - 4.91e3T^{2}
19 1+139.T+6.85e3T2 1 + 139.T + 6.85e3T^{2}
23 1−111.iT−1.21e4T2 1 - 111. iT - 1.21e4T^{2}
29 1−24.9T+2.43e4T2 1 - 24.9T + 2.43e4T^{2}
31 1−31.4T+2.97e4T2 1 - 31.4T + 2.97e4T^{2}
37 1−13.1iT−5.06e4T2 1 - 13.1iT - 5.06e4T^{2}
41 1−261.T+6.89e4T2 1 - 261.T + 6.89e4T^{2}
43 1−57.7iT−7.95e4T2 1 - 57.7iT - 7.95e4T^{2}
47 1+343.iT−1.03e5T2 1 + 343. iT - 1.03e5T^{2}
53 1−342.iT−1.48e5T2 1 - 342. iT - 1.48e5T^{2}
59 1+88.3T+2.05e5T2 1 + 88.3T + 2.05e5T^{2}
61 1−738.T+2.26e5T2 1 - 738.T + 2.26e5T^{2}
67 1−342.iT−3.00e5T2 1 - 342. iT - 3.00e5T^{2}
71 1+207.T+3.57e5T2 1 + 207.T + 3.57e5T^{2}
73 1−1.01e3iT−3.89e5T2 1 - 1.01e3iT - 3.89e5T^{2}
79 1+1.29e3T+4.93e5T2 1 + 1.29e3T + 4.93e5T^{2}
83 1+441.iT−5.71e5T2 1 + 441. iT - 5.71e5T^{2}
89 1−1.48e3T+7.04e5T2 1 - 1.48e3T + 7.04e5T^{2}
97 1−1.34e3iT−9.12e5T2 1 - 1.34e3iT - 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

−11.80149489799244784035021206805, −10.94558535827759543346730258278, −10.19800569759812552753653228330, −9.117106124860681849433193532415, −8.332680359209660022192099951077, −7.15713222415703252182524346391, −5.93541461466281903632609737641, −5.10930925873633380741585036787, −4.05790502153425892943474322216, −2.56289422630260238368321333213, 0.50684220798334231562226717325, 1.80197377956965645335416910917, 2.64246023915408329191107017449, 4.20654532501381803700741064982, 6.13244279567184014590871469775, 6.78465561241106532973337912711, 7.81772965597152635372328368448, 8.794735396771399739850685654308, 10.25068997804125113722272531850, 10.95629529726510118124544253037

Graph of the ZZ-function along the critical line