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Properties

Label 2-1815-1.1-c3-0-166
Degree 22
Conductor 18151815
Sign 11
Analytic cond. 107.088107.088
Root an. cond. 10.348310.3483
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.26·2-s + 3·3-s + 19.6·4-s + 5·5-s + 15.7·6-s − 5.37·7-s + 61.4·8-s + 9·9-s + 26.3·10-s + 59.0·12-s + 5.15·13-s − 28.2·14-s + 15·15-s + 166.·16-s − 107.·17-s + 47.3·18-s + 70.1·19-s + 98.4·20-s − 16.1·21-s + 104.·23-s + 184.·24-s + 25·25-s + 27.1·26-s + 27·27-s − 105.·28-s + 122.·29-s + 78.9·30-s + ⋯
L(s)  = 1  + 1.86·2-s + 0.577·3-s + 2.46·4-s + 0.447·5-s + 1.07·6-s − 0.290·7-s + 2.71·8-s + 0.333·9-s + 0.831·10-s + 1.42·12-s + 0.109·13-s − 0.539·14-s + 0.258·15-s + 2.59·16-s − 1.53·17-s + 0.620·18-s + 0.847·19-s + 1.10·20-s − 0.167·21-s + 0.947·23-s + 1.56·24-s + 0.200·25-s + 0.204·26-s + 0.192·27-s − 0.713·28-s + 0.784·29-s + 0.480·30-s + ⋯

Functional equation

Λ(s)=(1815s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1815s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18151815    =    3⋅5⋅1123 \cdot 5 \cdot 11^{2}
Sign: 11
Analytic conductor: 107.088107.088
Root analytic conductor: 10.348310.3483
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1815, ( :3/2), 1)(2,\ 1815,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 10.9374674910.93746749
L(12)L(\frac12) ≈\approx 10.9374674910.93746749
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)
bad3 1−3T 1 - 3T
5 1−5T 1 - 5T
11 1 1
good2 1−5.26T+8T2 1 - 5.26T + 8T^{2}
7 1+5.37T+343T2 1 + 5.37T + 343T^{2}
13 1−5.15T+2.19e3T2 1 - 5.15T + 2.19e3T^{2}
17 1+107.T+4.91e3T2 1 + 107.T + 4.91e3T^{2}
19 1−70.1T+6.85e3T2 1 - 70.1T + 6.85e3T^{2}
23 1−104.T+1.21e4T2 1 - 104.T + 1.21e4T^{2}
29 1−122.T+2.43e4T2 1 - 122.T + 2.43e4T^{2}
31 1−252.T+2.97e4T2 1 - 252.T + 2.97e4T^{2}
37 1−150.T+5.06e4T2 1 - 150.T + 5.06e4T^{2}
41 1−17.3T+6.89e4T2 1 - 17.3T + 6.89e4T^{2}
43 1−459.T+7.95e4T2 1 - 459.T + 7.95e4T^{2}
47 1−310.T+1.03e5T2 1 - 310.T + 1.03e5T^{2}
53 1+322.T+1.48e5T2 1 + 322.T + 1.48e5T^{2}
59 1−425.T+2.05e5T2 1 - 425.T + 2.05e5T^{2}
61 1+75.2T+2.26e5T2 1 + 75.2T + 2.26e5T^{2}
67 1+898.T+3.00e5T2 1 + 898.T + 3.00e5T^{2}
71 1−1.19e3T+3.57e5T2 1 - 1.19e3T + 3.57e5T^{2}
73 1+277.T+3.89e5T2 1 + 277.T + 3.89e5T^{2}
79 1+1.25e3T+4.93e5T2 1 + 1.25e3T + 4.93e5T^{2}
83 1−156.T+5.71e5T2 1 - 156.T + 5.71e5T^{2}
89 1+96.0T+7.04e5T2 1 + 96.0T + 7.04e5T^{2}
97 1+840.T+9.12e5T2 1 + 840.T + 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

−8.913756565237369572990137677964, −7.85444138206706893240258514378, −6.89946616569584366996095580564, −6.43991654416683018619593579753, −5.54557555296018949605193382116, −4.65373309934304937388015097981, −4.06578856020299078175997031077, −2.87830680894750883760537362042, −2.55413511531942314166241788734, −1.25566548165062588526259291134, 1.25566548165062588526259291134, 2.55413511531942314166241788734, 2.87830680894750883760537362042, 4.06578856020299078175997031077, 4.65373309934304937388015097981, 5.54557555296018949605193382116, 6.43991654416683018619593579753, 6.89946616569584366996095580564, 7.85444138206706893240258514378, 8.913756565237369572990137677964

Graph of the ZZ-function along the critical line