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Properties

Label 2-2475-1.1-c3-0-21
Degree 22
Conductor 24752475
Sign 11
Analytic cond. 146.029146.029
Root an. cond. 12.084212.0842
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 + 22.9·4-s − 6.05·7-s − 83.0·8-s + 11·11-s + 4.38·13-s + 33.6·14-s + 278.·16-s − 110.·17-s − 94.2·19-s − 61.1·22-s + 15.7·23-s − 24.3·26-s − 138.·28-s + 256.·29-s − 170.·31-s − 883.·32-s + 614.·34-s + 190.·37-s + 524.·38-s − 249.·41-s − 291.·43-s + 252.·44-s − 87.6·46-s + 182.·47-s − 306.·49-s + 100.·52-s + ⋯
L(s)  = 1  − 1.96·2-s + 2.86·4-s − 0.326·7-s − 3.66·8-s + 0.301·11-s + 0.0935·13-s + 0.642·14-s + 4.34·16-s − 1.57·17-s − 1.13·19-s − 0.592·22-s + 0.142·23-s − 0.183·26-s − 0.936·28-s + 1.64·29-s − 0.988·31-s − 4.88·32-s + 3.10·34-s + 0.848·37-s + 2.23·38-s − 0.950·41-s − 1.03·43-s + 0.864·44-s − 0.280·46-s + 0.565·47-s − 0.893·49-s + 0.268·52-s + ⋯

Functional equation

Λ(s)=(2475s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2475s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24752475    =    32⋅52⋅113^{2} \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 146.029146.029
Root analytic conductor: 12.084212.0842
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2475, ( :3/2), 1)(2,\ 2475,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.40670648850.4067064885
L(12)L(\frac12) ≈\approx 0.40670648850.4067064885
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 1
5 1 1
11 1−11T 1 - 11T
good2 1+5.56T+8T2 1 + 5.56T + 8T^{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

−8.705511863156371384190229700646, −8.108192799794567087244957511553, −7.13581827526283837654995668023, −6.55434574309393826552331503202, −6.07320612588262219219533089877, −4.60383467572379345492065702527, −3.29025544063504161326413377142, −2.35675632314704089251216080042, −1.56671103981907851463601270980, −0.37525469839682219589408221175, 0.37525469839682219589408221175, 1.56671103981907851463601270980, 2.35675632314704089251216080042, 3.29025544063504161326413377142, 4.60383467572379345492065702527, 6.07320612588262219219533089877, 6.55434574309393826552331503202, 7.13581827526283837654995668023, 8.108192799794567087244957511553, 8.705511863156371384190229700646

Graph of the ZZ-function along the critical line