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Properties

Label 2-1254-1.1-c1-0-15
Degree 22
Conductor 12541254
Sign 11
Analytic cond. 10.013210.0132
Root an. cond. 3.164373.16437
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 4·13-s − 2·14-s + 15-s + 16-s + 3·17-s + 18-s − 19-s + 20-s − 2·21-s + 22-s + 4·23-s + 24-s − 4·25-s + 4·26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.436·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.784·26-s + 0.192·27-s − 0.377·28-s + ⋯

Functional equation

Λ(s)=(1254s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1254s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12541254    =    2⋅3⋅11⋅192 \cdot 3 \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 10.013210.0132
Root analytic conductor: 3.164373.16437
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1254, ( :1/2), 1)(2,\ 1254,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3918372193.391837219
L(12)L(\frac12) ≈\approx 3.3918372193.391837219
L(32)L(\frac{3}{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)
bad2 1−T 1 - T
3 1−T 1 - T
11 1−T 1 - T
19 1+T 1 + T
good5 1−T+pT2 1 - T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−9T+pT2 1 - 9 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+13T+pT2 1 + 13 T + p T^{2}
67 1−13T+pT2 1 - 13 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+16T+pT2 1 + 16 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+5T+pT2 1 + 5 T + p T^{2}
97 1−8T+pT2 1 - 8 T + p T^{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

−9.581944822888227403253424096380, −9.043718502561435427567016078815, −7.980179118153069516022158507229, −7.14273558884352157405370540959, −6.17372340837701233463853408340, −5.67999357501247308290616366784, −4.33257580321924699251330751557, −3.53794942184716470982502852668, −2.68938573736263700447372124696, −1.39093487590051632703460554838, 1.39093487590051632703460554838, 2.68938573736263700447372124696, 3.53794942184716470982502852668, 4.33257580321924699251330751557, 5.67999357501247308290616366784, 6.17372340837701233463853408340, 7.14273558884352157405370540959, 7.980179118153069516022158507229, 9.043718502561435427567016078815, 9.581944822888227403253424096380

Graph of the ZZ-function along the critical line