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Properties

Label 2-275-1.1-c3-0-8
Degree 22
Conductor 275275
Sign 11
Analytic cond. 16.225516.2255
Root an. cond. 4.028094.02809
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  − 4.89·2-s − 8.92·3-s + 15.9·4-s + 43.6·6-s + 17.6·7-s − 39.0·8-s + 52.6·9-s + 11·11-s − 142.·12-s + 43.3·13-s − 86.4·14-s + 63.2·16-s + 53.1·17-s − 257.·18-s − 75.6·19-s − 157.·21-s − 53.8·22-s − 181.·23-s + 348.·24-s − 212.·26-s − 228.·27-s + 282.·28-s − 23.8·29-s + 173.·31-s + 2.59·32-s − 98.1·33-s − 260.·34-s + ⋯
L(s)  = 1  − 1.73·2-s − 1.71·3-s + 1.99·4-s + 2.97·6-s + 0.953·7-s − 1.72·8-s + 1.94·9-s + 0.301·11-s − 3.42·12-s + 0.924·13-s − 1.65·14-s + 0.987·16-s + 0.758·17-s − 3.37·18-s − 0.913·19-s − 1.63·21-s − 0.521·22-s − 1.64·23-s + 2.96·24-s − 1.59·26-s − 1.63·27-s + 1.90·28-s − 0.153·29-s + 1.00·31-s + 0.0143·32-s − 0.517·33-s − 1.31·34-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: 11
Analytic conductor: 16.225516.2255
Root analytic conductor: 4.028094.02809
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 275, ( :3/2), 1)(2,\ 275,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.45663745190.4566374519
L(12)L(\frac12) ≈\approx 0.45663745190.4566374519
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)
bad5 1 1
11 1−11T 1 - 11T
good2 1+4.89T+8T2 1 + 4.89T + 8T^{2}
3 1+8.92T+27T2 1 + 8.92T + 27T^{2}
7 1−17.6T+343T2 1 - 17.6T + 343T^{2}
13 1−43.3T+2.19e3T2 1 - 43.3T + 2.19e3T^{2}
17 1−53.1T+4.91e3T2 1 - 53.1T + 4.91e3T^{2}
19 1+75.6T+6.85e3T2 1 + 75.6T + 6.85e3T^{2}
23 1+181.T+1.21e4T2 1 + 181.T + 1.21e4T^{2}
29 1+23.8T+2.43e4T2 1 + 23.8T + 2.43e4T^{2}
31 1−173.T+2.97e4T2 1 - 173.T + 2.97e4T^{2}
37 1−239.T+5.06e4T2 1 - 239.T + 5.06e4T^{2}
41 1+56.9T+6.89e4T2 1 + 56.9T + 6.89e4T^{2}
43 1−334.T+7.95e4T2 1 - 334.T + 7.95e4T^{2}
47 1+381.T+1.03e5T2 1 + 381.T + 1.03e5T^{2}
53 1+187.T+1.48e5T2 1 + 187.T + 1.48e5T^{2}
59 1−369.T+2.05e5T2 1 - 369.T + 2.05e5T^{2}
61 1−361.T+2.26e5T2 1 - 361.T + 2.26e5T^{2}
67 1−267.T+3.00e5T2 1 - 267.T + 3.00e5T^{2}
71 1−94.9T+3.57e5T2 1 - 94.9T + 3.57e5T^{2}
73 1+429.T+3.89e5T2 1 + 429.T + 3.89e5T^{2}
79 1+230.T+4.93e5T2 1 + 230.T + 4.93e5T^{2}
83 1+1.31e3T+5.71e5T2 1 + 1.31e3T + 5.71e5T^{2}
89 1−272.T+7.04e5T2 1 - 272.T + 7.04e5T^{2}
97 1−100.T+9.12e5T2 1 - 100.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

−11.41175190214076318652996487190, −10.45836313072359570918933452185, −9.854981104525944265769995063058, −8.487858840218890432582919125641, −7.73302278076985291278529148830, −6.52864226420140714098746122902, −5.84199059987235508841036261076, −4.38166770879649062919849233139, −1.74067982748143129091184094433, −0.70842764654648443474504148642, 0.70842764654648443474504148642, 1.74067982748143129091184094433, 4.38166770879649062919849233139, 5.84199059987235508841036261076, 6.52864226420140714098746122902, 7.73302278076985291278529148830, 8.487858840218890432582919125641, 9.854981104525944265769995063058, 10.45836313072359570918933452185, 11.41175190214076318652996487190

Graph of the ZZ-function along the critical line