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Properties

Label 2-90e2-5.4-c1-0-55
Degree 22
Conductor 81008100
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 64.678864.6788
Root an. cond. 8.042318.04231
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  + 4i·7-s + 3·11-s − 4i·13-s − 5·19-s − 6i·23-s + 9·29-s + 5·31-s − 2i·37-s − 9·41-s − 10i·43-s + 6i·47-s − 9·49-s − 12i·53-s − 9·59-s − 10·61-s + ⋯
L(s)  = 1  + 1.51i·7-s + 0.904·11-s − 1.10i·13-s − 1.14·19-s − 1.25i·23-s + 1.67·29-s + 0.898·31-s − 0.328i·37-s − 1.40·41-s − 1.52i·43-s + 0.875i·47-s − 1.28·49-s − 1.64i·53-s − 1.17·59-s − 1.28·61-s + ⋯

Functional equation

Λ(s)=(8100s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(8100s/2 ΓC(s+1/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 81008100    =    22⋅34⋅522^{2} \cdot 3^{4} \cdot 5^{2}
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 64.678864.6788
Root analytic conductor: 8.042318.04231
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ8100(649,⋅)\chi_{8100} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 8100, ( :1/2), 0.447+0.894i)(2,\ 8100,\ (\ :1/2),\ 0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.5800756071.580075607
L(12)L(\frac12) ≈\approx 1.5800756071.580075607
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)
bad2 1 1
3 1 1
5 1 1
good7 1−4iT−7T2 1 - 4iT - 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1−17T2 1 - 17T^{2}
19 1+5T+19T2 1 + 5T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1−9T+29T2 1 - 9T + 29T^{2}
31 1−5T+31T2 1 - 5T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+9T+41T2 1 + 9T + 41T^{2}
43 1+10iT−43T2 1 + 10iT - 43T^{2}
47 1−6iT−47T2 1 - 6iT - 47T^{2}
53 1+12iT−53T2 1 + 12iT - 53T^{2}
59 1+9T+59T2 1 + 9T + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1+2iT−67T2 1 + 2iT - 67T^{2}
71 1−3T+71T2 1 - 3T + 71T^{2}
73 1+4iT−73T2 1 + 4iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−6iT−83T2 1 - 6iT - 83T^{2}
89 1−9T+89T2 1 - 9T + 89T^{2}
97 1+2iT−97T2 1 + 2iT - 97T^{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

−8.014314369223243594461048982039, −6.60597526108958956137347048781, −6.51472452494962458507393716037, −5.64383531484979410889158488991, −4.95059647334067404023635441222, −4.24118536048522755757380064591, −3.16774684067533091896441644117, −2.57730295356432369557971685450, −1.71680167446038212206007107819, −0.39933647876750581816733663898, 1.05487103233014165842616105893, 1.67560496926417688775667522351, 2.94407829053737253396460231590, 3.81109860777580603617551319637, 4.38374202551815685015196066995, 4.85091576748896371807942654703, 6.23906877904912093020757912163, 6.53239127785860156602342407887, 7.16367256370461133833970837946, 7.896991764601046083267678194026

Graph of the ZZ-function along the critical line