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Properties

Label 2-275-1.1-c3-0-36
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  + 5.56·2-s + 3.56·3-s + 22.9·4-s + 19.8·6-s − 6.05·7-s + 83.0·8-s − 14.3·9-s − 11·11-s + 81.6·12-s + 4.38·13-s − 33.6·14-s + 278.·16-s + 110.·17-s − 79.6·18-s − 94.2·19-s − 21.5·21-s − 61.1·22-s − 15.7·23-s + 295.·24-s + 24.3·26-s − 147.·27-s − 138.·28-s − 256.·29-s − 170.·31-s + 883.·32-s − 39.1·33-s + 614.·34-s + ⋯
L(s)  = 1  + 1.96·2-s + 0.685·3-s + 2.86·4-s + 1.34·6-s − 0.326·7-s + 3.66·8-s − 0.530·9-s − 0.301·11-s + 1.96·12-s + 0.0935·13-s − 0.642·14-s + 4.34·16-s + 1.57·17-s − 1.04·18-s − 1.13·19-s − 0.224·21-s − 0.592·22-s − 0.142·23-s + 2.51·24-s + 0.183·26-s − 1.04·27-s − 0.936·28-s − 1.64·29-s − 0.988·31-s + 4.88·32-s − 0.206·33-s + 3.10·34-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 7.1474023787.147402378
L(12)L(\frac12) ≈\approx 7.1474023787.147402378
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 1−5.56T+8T2 1 - 5.56T + 8T^{2}
3 1−3.56T+27T2 1 - 3.56T + 27T^{2}
7 1+6.05T+343T2 1 + 6.05T + 343T^{2}
13 1−4.38T+2.19e3T2 1 - 4.38T + 2.19e3T^{2}
17 1−110.T+4.91e3T2 1 - 110.T + 4.91e3T^{2}
19 1+94.2T+6.85e3T2 1 + 94.2T + 6.85e3T^{2}
23 1+15.7T+1.21e4T2 1 + 15.7T + 1.21e4T^{2}
29 1+256.T+2.43e4T2 1 + 256.T + 2.43e4T^{2}
31 1+170.T+2.97e4T2 1 + 170.T + 2.97e4T^{2}
37 1−190.T+5.06e4T2 1 - 190.T + 5.06e4T^{2}
41 1−249.T+6.89e4T2 1 - 249.T + 6.89e4T^{2}
43 1+291.T+7.95e4T2 1 + 291.T + 7.95e4T^{2}
47 1+182.T+1.03e5T2 1 + 182.T + 1.03e5T^{2}
53 1−289.T+1.48e5T2 1 - 289.T + 1.48e5T^{2}
59 1−282.T+2.05e5T2 1 - 282.T + 2.05e5T^{2}
61 1−167.T+2.26e5T2 1 - 167.T + 2.26e5T^{2}
67 1−176.T+3.00e5T2 1 - 176.T + 3.00e5T^{2}
71 1−919.T+3.57e5T2 1 - 919.T + 3.57e5T^{2}
73 1+154.T+3.89e5T2 1 + 154.T + 3.89e5T^{2}
79 1+882.T+4.93e5T2 1 + 882.T + 4.93e5T^{2}
83 1+277.T+5.71e5T2 1 + 277.T + 5.71e5T^{2}
89 1+977.T+7.04e5T2 1 + 977.T + 7.04e5T^{2}
97 1−1.10e3T+9.12e5T2 1 - 1.10e3T + 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

−11.69572016605328762640258971878, −10.94642779717614699350595196286, −9.791117252549633231185960275416, −8.178086329002355171538578448866, −7.29460882652785275914456489596, −6.05652779798234456546893902156, −5.34495203944260822115612825842, −3.93093175354559237603576379166, −3.15864310528863240940297641906, −2.03656860698502110908573076507, 2.03656860698502110908573076507, 3.15864310528863240940297641906, 3.93093175354559237603576379166, 5.34495203944260822115612825842, 6.05652779798234456546893902156, 7.29460882652785275914456489596, 8.178086329002355171538578448866, 9.791117252549633231185960275416, 10.94642779717614699350595196286, 11.69572016605328762640258971878

Graph of the ZZ-function along the critical line