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Properties

Label 2-2175-1.1-c3-0-90
Degree 22
Conductor 21752175
Sign 11
Analytic cond. 128.329128.329
Root an. cond. 11.328211.3282
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  − 0.450·2-s − 3·3-s − 7.79·4-s + 1.35·6-s − 0.788·7-s + 7.11·8-s + 9·9-s + 67.3·11-s + 23.3·12-s − 28.5·13-s + 0.355·14-s + 59.1·16-s + 10.0·17-s − 4.05·18-s + 149.·19-s + 2.36·21-s − 30.3·22-s + 2.58·23-s − 21.3·24-s + 12.8·26-s − 27·27-s + 6.15·28-s − 29·29-s + 114.·31-s − 83.6·32-s − 202.·33-s − 4.51·34-s + ⋯
L(s)  = 1  − 0.159·2-s − 0.577·3-s − 0.974·4-s + 0.0919·6-s − 0.0425·7-s + 0.314·8-s + 0.333·9-s + 1.84·11-s + 0.562·12-s − 0.608·13-s + 0.00678·14-s + 0.924·16-s + 0.142·17-s − 0.0531·18-s + 1.79·19-s + 0.0245·21-s − 0.294·22-s + 0.0234·23-s − 0.181·24-s + 0.0969·26-s − 0.192·27-s + 0.0415·28-s − 0.185·29-s + 0.664·31-s − 0.461·32-s − 1.06·33-s − 0.0227·34-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 11
Analytic conductor: 128.329128.329
Root analytic conductor: 11.328211.3282
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2175, ( :3/2), 1)(2,\ 2175,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.6163637451.616363745
L(12)L(\frac12) ≈\approx 1.6163637451.616363745
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 1
29 1+29T 1 + 29T
good2 1+0.450T+8T2 1 + 0.450T + 8T^{2}
7 1+0.788T+343T2 1 + 0.788T + 343T^{2}
11 1−67.3T+1.33e3T2 1 - 67.3T + 1.33e3T^{2}
13 1+28.5T+2.19e3T2 1 + 28.5T + 2.19e3T^{2}
17 1−10.0T+4.91e3T2 1 - 10.0T + 4.91e3T^{2}
19 1−149.T+6.85e3T2 1 - 149.T + 6.85e3T^{2}
23 1−2.58T+1.21e4T2 1 - 2.58T + 1.21e4T^{2}
31 1−114.T+2.97e4T2 1 - 114.T + 2.97e4T^{2}
37 1−399.T+5.06e4T2 1 - 399.T + 5.06e4T^{2}
41 1−340.T+6.89e4T2 1 - 340.T + 6.89e4T^{2}
43 1−242.T+7.95e4T2 1 - 242.T + 7.95e4T^{2}
47 1−376.T+1.03e5T2 1 - 376.T + 1.03e5T^{2}
53 1+263.T+1.48e5T2 1 + 263.T + 1.48e5T^{2}
59 1−395.T+2.05e5T2 1 - 395.T + 2.05e5T^{2}
61 1+541.T+2.26e5T2 1 + 541.T + 2.26e5T^{2}
67 1+496.T+3.00e5T2 1 + 496.T + 3.00e5T^{2}
71 1+81.1T+3.57e5T2 1 + 81.1T + 3.57e5T^{2}
73 1+1.04e3T+3.89e5T2 1 + 1.04e3T + 3.89e5T^{2}
79 1−153.T+4.93e5T2 1 - 153.T + 4.93e5T^{2}
83 1−398.T+5.71e5T2 1 - 398.T + 5.71e5T^{2}
89 1+362.T+7.04e5T2 1 + 362.T + 7.04e5T^{2}
97 1+402.T+9.12e5T2 1 + 402.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

−9.016131290186121865618564993472, −7.83807330737388434799698371361, −7.28102594063034339569483808982, −6.22519449546332642274677670019, −5.59197106295520307364695039111, −4.57829112296476850219372112952, −4.07258750668923793812241983742, −2.99697460317596903302210651314, −1.33978268393279870572231352633, −0.71770530559250330228077940596, 0.71770530559250330228077940596, 1.33978268393279870572231352633, 2.99697460317596903302210651314, 4.07258750668923793812241983742, 4.57829112296476850219372112952, 5.59197106295520307364695039111, 6.22519449546332642274677670019, 7.28102594063034339569483808982, 7.83807330737388434799698371361, 9.016131290186121865618564993472

Graph of the ZZ-function along the critical line