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Properties

Label 2-165-1.1-c3-0-9
Degree 22
Conductor 165165
Sign −1-1
Analytic cond. 9.735319.73531
Root an. cond. 3.120143.12014
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.32·2-s − 3·3-s − 2.57·4-s − 5·5-s + 6.98·6-s + 22.4·7-s + 24.6·8-s + 9·9-s + 11.6·10-s + 11·11-s + 7.72·12-s − 9.86·13-s − 52.3·14-s + 15·15-s − 36.7·16-s − 128.·17-s − 20.9·18-s + 7.04·19-s + 12.8·20-s − 67.4·21-s − 25.6·22-s + 0.654·23-s − 73.8·24-s + 25·25-s + 22.9·26-s − 27·27-s − 57.8·28-s + ⋯
L(s)  = 1  − 0.823·2-s − 0.577·3-s − 0.321·4-s − 0.447·5-s + 0.475·6-s + 1.21·7-s + 1.08·8-s + 0.333·9-s + 0.368·10-s + 0.301·11-s + 0.185·12-s − 0.210·13-s − 0.998·14-s + 0.258·15-s − 0.574·16-s − 1.82·17-s − 0.274·18-s + 0.0850·19-s + 0.143·20-s − 0.700·21-s − 0.248·22-s + 0.00593·23-s − 0.628·24-s + 0.200·25-s + 0.173·26-s − 0.192·27-s − 0.390·28-s + ⋯

Functional equation

Λ(s)=(165s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(165s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 165165    =    3⋅5⋅113 \cdot 5 \cdot 11
Sign: −1-1
Analytic conductor: 9.735319.73531
Root analytic conductor: 3.120143.12014
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 165, ( :3/2), −1)(2,\ 165,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1−11T 1 - 11T
good2 1+2.32T+8T2 1 + 2.32T + 8T^{2}
7 1−22.4T+343T2 1 - 22.4T + 343T^{2}
13 1+9.86T+2.19e3T2 1 + 9.86T + 2.19e3T^{2}
17 1+128.T+4.91e3T2 1 + 128.T + 4.91e3T^{2}
19 1−7.04T+6.85e3T2 1 - 7.04T + 6.85e3T^{2}
23 1−0.654T+1.21e4T2 1 - 0.654T + 1.21e4T^{2}
29 1+229.T+2.43e4T2 1 + 229.T + 2.43e4T^{2}
31 1−155.T+2.97e4T2 1 - 155.T + 2.97e4T^{2}
37 1+110.T+5.06e4T2 1 + 110.T + 5.06e4T^{2}
41 1−154.T+6.89e4T2 1 - 154.T + 6.89e4T^{2}
43 1+401.T+7.95e4T2 1 + 401.T + 7.95e4T^{2}
47 1+277.T+1.03e5T2 1 + 277.T + 1.03e5T^{2}
53 1+651.T+1.48e5T2 1 + 651.T + 1.48e5T^{2}
59 1+423.T+2.05e5T2 1 + 423.T + 2.05e5T^{2}
61 1−681.T+2.26e5T2 1 - 681.T + 2.26e5T^{2}
67 1−374.T+3.00e5T2 1 - 374.T + 3.00e5T^{2}
71 1−96.6T+3.57e5T2 1 - 96.6T + 3.57e5T^{2}
73 1+19.9T+3.89e5T2 1 + 19.9T + 3.89e5T^{2}
79 1−24.4T+4.93e5T2 1 - 24.4T + 4.93e5T^{2}
83 1+1.12e3T+5.71e5T2 1 + 1.12e3T + 5.71e5T^{2}
89 1+639.T+7.04e5T2 1 + 639.T + 7.04e5T^{2}
97 1+730.T+9.12e5T2 1 + 730.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

−11.38396107918053199134896155671, −11.06426070673374402137520530015, −9.784012941071971075327506411725, −8.712145943629755070877300170293, −7.901200733731107632500869835234, −6.78837948449091974826726938817, −5.05832509076758630826783281233, −4.23912785895003769931341466511, −1.66260061704225131346789765018, 0, 1.66260061704225131346789765018, 4.23912785895003769931341466511, 5.05832509076758630826783281233, 6.78837948449091974826726938817, 7.901200733731107632500869835234, 8.712145943629755070877300170293, 9.784012941071971075327506411725, 11.06426070673374402137520530015, 11.38396107918053199134896155671

Graph of the ZZ-function along the critical line