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Properties

Label 2-1815-1.1-c3-0-90
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  − 1.82·2-s + 3·3-s − 4.65·4-s + 5·5-s − 5.48·6-s + 15.0·7-s + 23.1·8-s + 9·9-s − 9.13·10-s − 13.9·12-s + 11.4·13-s − 27.5·14-s + 15·15-s − 5.02·16-s − 11.9·17-s − 16.4·18-s + 91.0·19-s − 23.2·20-s + 45.2·21-s + 103.·23-s + 69.4·24-s + 25·25-s − 20.9·26-s + 27·27-s − 70.3·28-s + 99.3·29-s − 27.4·30-s + ⋯
L(s)  = 1  − 0.646·2-s + 0.577·3-s − 0.582·4-s + 0.447·5-s − 0.373·6-s + 0.814·7-s + 1.02·8-s + 0.333·9-s − 0.289·10-s − 0.336·12-s + 0.243·13-s − 0.526·14-s + 0.258·15-s − 0.0785·16-s − 0.170·17-s − 0.215·18-s + 1.09·19-s − 0.260·20-s + 0.470·21-s + 0.935·23-s + 0.590·24-s + 0.200·25-s − 0.157·26-s + 0.192·27-s − 0.474·28-s + 0.636·29-s − 0.166·30-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 2.3161585552.316158555
L(12)L(\frac12) ≈\approx 2.3161585552.316158555
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+1.82T+8T2 1 + 1.82T + 8T^{2}
7 1−15.0T+343T2 1 - 15.0T + 343T^{2}
13 1−11.4T+2.19e3T2 1 - 11.4T + 2.19e3T^{2}
17 1+11.9T+4.91e3T2 1 + 11.9T + 4.91e3T^{2}
19 1−91.0T+6.85e3T2 1 - 91.0T + 6.85e3T^{2}
23 1−103.T+1.21e4T2 1 - 103.T + 1.21e4T^{2}
29 1−99.3T+2.43e4T2 1 - 99.3T + 2.43e4T^{2}
31 1+39.9T+2.97e4T2 1 + 39.9T + 2.97e4T^{2}
37 1−149.T+5.06e4T2 1 - 149.T + 5.06e4T^{2}
41 1−135.T+6.89e4T2 1 - 135.T + 6.89e4T^{2}
43 1+304.T+7.95e4T2 1 + 304.T + 7.95e4T^{2}
47 1−113.T+1.03e5T2 1 - 113.T + 1.03e5T^{2}
53 1−552.T+1.48e5T2 1 - 552.T + 1.48e5T^{2}
59 1+251.T+2.05e5T2 1 + 251.T + 2.05e5T^{2}
61 1−211.T+2.26e5T2 1 - 211.T + 2.26e5T^{2}
67 1−947.T+3.00e5T2 1 - 947.T + 3.00e5T^{2}
71 1+175.T+3.57e5T2 1 + 175.T + 3.57e5T^{2}
73 1+768.T+3.89e5T2 1 + 768.T + 3.89e5T^{2}
79 1−710.T+4.93e5T2 1 - 710.T + 4.93e5T^{2}
83 1+902.T+5.71e5T2 1 + 902.T + 5.71e5T^{2}
89 1+482.T+7.04e5T2 1 + 482.T + 7.04e5T^{2}
97 1−724.T+9.12e5T2 1 - 724.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

−8.890083364805225707599983787068, −8.277628632194984276095488158850, −7.59872927208291704091565699168, −6.79802891093884596830712156232, −5.49568455765400559022869348576, −4.84314368646538753197546411141, −3.93732599849672536407076968952, −2.80037734563663105944052343802, −1.61923964591889839547319110050, −0.844883549829202385190923874806, 0.844883549829202385190923874806, 1.61923964591889839547319110050, 2.80037734563663105944052343802, 3.93732599849672536407076968952, 4.84314368646538753197546411141, 5.49568455765400559022869348576, 6.79802891093884596830712156232, 7.59872927208291704091565699168, 8.277628632194984276095488158850, 8.890083364805225707599983787068

Graph of the ZZ-function along the critical line