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Properties

Label 2-275-5.4-c3-0-1
Degree 22
Conductor 275275
Sign 0.894+0.447i0.894 + 0.447i
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  + 4.72i·2-s + 8.29i·3-s − 14.2·4-s − 39.1·6-s + 5.48i·7-s − 29.6i·8-s − 41.8·9-s − 11·11-s − 118. i·12-s − 84.9i·13-s − 25.8·14-s + 25.8·16-s + 119. i·17-s − 197. i·18-s + 54.4·19-s + ⋯
L(s)  = 1  + 1.66i·2-s + 1.59i·3-s − 1.78·4-s − 2.66·6-s + 0.296i·7-s − 1.31i·8-s − 1.54·9-s − 0.301·11-s − 2.85i·12-s − 1.81i·13-s − 0.494·14-s + 0.403·16-s + 1.70i·17-s − 2.58i·18-s + 0.657·19-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: 0.894+0.447i0.894 + 0.447i
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), 0.894+0.447i)(2,\ 275,\ (\ :3/2),\ 0.894 + 0.447i)

Particular Values

L(2)L(2) ≈\approx 0.67590779200.6759077920
L(12)L(\frac12) ≈\approx 0.67590779200.6759077920
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 1−4.72iT−8T2 1 - 4.72iT - 8T^{2}
3 1−8.29iT−27T2 1 - 8.29iT - 27T^{2}
7 1−5.48iT−343T2 1 - 5.48iT - 343T^{2}
13 1+84.9iT−2.19e3T2 1 + 84.9iT - 2.19e3T^{2}
17 1−119.iT−4.91e3T2 1 - 119. iT - 4.91e3T^{2}
19 1−54.4T+6.85e3T2 1 - 54.4T + 6.85e3T^{2}
23 1+83.8iT−1.21e4T2 1 + 83.8iT - 1.21e4T^{2}
29 1+296.T+2.43e4T2 1 + 296.T + 2.43e4T^{2}
31 1+18.7T+2.97e4T2 1 + 18.7T + 2.97e4T^{2}
37 1−188.iT−5.06e4T2 1 - 188. iT - 5.06e4T^{2}
41 1+209.T+6.89e4T2 1 + 209.T + 6.89e4T^{2}
43 1−183.iT−7.95e4T2 1 - 183. iT - 7.95e4T^{2}
47 1−142.iT−1.03e5T2 1 - 142. iT - 1.03e5T^{2}
53 1−610.iT−1.48e5T2 1 - 610. iT - 1.48e5T^{2}
59 1+657.T+2.05e5T2 1 + 657.T + 2.05e5T^{2}
61 1−171.T+2.26e5T2 1 - 171.T + 2.26e5T^{2}
67 1−116.iT−3.00e5T2 1 - 116. iT - 3.00e5T^{2}
71 1+595.T+3.57e5T2 1 + 595.T + 3.57e5T^{2}
73 1+629.iT−3.89e5T2 1 + 629. iT - 3.89e5T^{2}
79 1−935.T+4.93e5T2 1 - 935.T + 4.93e5T^{2}
83 1+81.2iT−5.71e5T2 1 + 81.2iT - 5.71e5T^{2}
89 1−245.T+7.04e5T2 1 - 245.T + 7.04e5T^{2}
97 1+395.iT−9.12e5T2 1 + 395. iT - 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

−12.57799916908343130269457437814, −10.92842778382003598494885953453, −10.24473706013545104077301983528, −9.257635684984383323043225627305, −8.389401812096744054446755482451, −7.64408028145699804128720535006, −5.99644043402791295616668810242, −5.48366245188070697856531022938, −4.48133110358615184364118860037, −3.29808537536948226357517800564, 0.26039580836544188718862138601, 1.54390704751961995439290636569, 2.36515163695079310186133518764, 3.72768400217465217086850630332, 5.22604095720427103494935853470, 6.86659324522835948518007764776, 7.51686081552272616642726848797, 9.002781477432539888605473244754, 9.617455278794651899312289930511, 11.06713279403074379276289708453

Graph of the ZZ-function along the critical line