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Properties

Label 2-2028-13.12-c3-0-22
Degree 22
Conductor 20282028
Sign −0.832−0.554i-0.832 - 0.554i
Analytic cond. 119.655119.655
Root an. cond. 10.938710.9387
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  + 3·3-s − 2i·5-s + 32i·7-s + 9·9-s + 68i·11-s − 6i·15-s + 14·17-s + 4i·19-s + 96i·21-s − 72·23-s + 121·25-s + 27·27-s + 102·29-s − 136i·31-s + 204i·33-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.178i·5-s + 1.72i·7-s + 0.333·9-s + 1.86i·11-s − 0.103i·15-s + 0.199·17-s + 0.0482i·19-s + 0.997i·21-s − 0.652·23-s + 0.967·25-s + 0.192·27-s + 0.653·29-s − 0.787i·31-s + 1.07i·33-s + ⋯

Functional equation

Λ(s)=(2028s/2 ΓC(s) L(s)=((−0.832−0.554i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(2028s/2 ΓC(s+3/2) L(s)=((−0.832−0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20282028    =    22⋅3⋅1322^{2} \cdot 3 \cdot 13^{2}
Sign: −0.832−0.554i-0.832 - 0.554i
Analytic conductor: 119.655119.655
Root analytic conductor: 10.938710.9387
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ2028(337,⋅)\chi_{2028} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2028, ( :3/2), −0.832−0.554i)(2,\ 2028,\ (\ :3/2),\ -0.832 - 0.554i)

Particular Values

L(2)L(2) ≈\approx 2.3075769872.307576987
L(12)L(\frac12) ≈\approx 2.3075769872.307576987
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)
bad2 1 1
3 1−3T 1 - 3T
13 1 1
good5 1+2iT−125T2 1 + 2iT - 125T^{2}
7 1−32iT−343T2 1 - 32iT - 343T^{2}
11 1−68iT−1.33e3T2 1 - 68iT - 1.33e3T^{2}
17 1−14T+4.91e3T2 1 - 14T + 4.91e3T^{2}
19 1−4iT−6.85e3T2 1 - 4iT - 6.85e3T^{2}
23 1+72T+1.21e4T2 1 + 72T + 1.21e4T^{2}
29 1−102T+2.43e4T2 1 - 102T + 2.43e4T^{2}
31 1+136iT−2.97e4T2 1 + 136iT - 2.97e4T^{2}
37 1−386iT−5.06e4T2 1 - 386iT - 5.06e4T^{2}
41 1−250iT−6.89e4T2 1 - 250iT - 6.89e4T^{2}
43 1−140T+7.95e4T2 1 - 140T + 7.95e4T^{2}
47 1−296iT−1.03e5T2 1 - 296iT - 1.03e5T^{2}
53 1−526T+1.48e5T2 1 - 526T + 1.48e5T^{2}
59 1+332iT−2.05e5T2 1 + 332iT - 2.05e5T^{2}
61 1+410T+2.26e5T2 1 + 410T + 2.26e5T^{2}
67 1−596iT−3.00e5T2 1 - 596iT - 3.00e5T^{2}
71 1+880iT−3.57e5T2 1 + 880iT - 3.57e5T^{2}
73 1+506iT−3.89e5T2 1 + 506iT - 3.89e5T^{2}
79 1+640T+4.93e5T2 1 + 640T + 4.93e5T^{2}
83 1−1.38e3iT−5.71e5T2 1 - 1.38e3iT - 5.71e5T^{2}
89 1+1.45e3iT−7.04e5T2 1 + 1.45e3iT - 7.04e5T^{2}
97 1+446iT−9.12e5T2 1 + 446iT - 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

−9.124274325822656895203433760919, −8.374353947812951283536540023160, −7.71823997685475186750354013703, −6.75685636226184684741923616383, −5.96066261953728101922559712026, −4.94688989206704124506213762350, −4.39768572170333406033630841599, −2.99794688425899016482321795487, −2.33425926609340014814110981864, −1.48225111171507079653476834222, 0.44891388009551541113233175732, 1.16898147348237885581706773418, 2.64924450538313261881176546796, 3.60408325007648737536345349268, 4.02445682722962573778310429542, 5.23672851518401069917398648869, 6.22415003086038462786123877335, 7.06620956274989615360234939962, 7.64628119322417793580509004451, 8.519926980992739635935754671017

Graph of the ZZ-function along the critical line