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Properties

Label 2-1254-1.1-c1-0-1
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 + 6-s − 4·7-s − 8-s + 9-s − 11-s − 12-s + 4·14-s + 16-s − 2·17-s − 18-s + 19-s + 4·21-s + 22-s − 2·23-s + 24-s − 5·25-s − 27-s − 4·28-s + 10·29-s − 2·31-s − 32-s + 33-s + 2·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.872·21-s + 0.213·22-s − 0.417·23-s + 0.204·24-s − 25-s − 0.192·27-s − 0.755·28-s + 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.174·33-s + 0.342·34-s + 1/6·36-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 0.60040916800.6004091680
L(12)L(\frac12) ≈\approx 0.60040916800.6004091680
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+pT2 1 + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−10T+pT2 1 - 10 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+16T+pT2 1 + 16 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−9.592472309668874924894894158066, −9.193634537030711191031976589522, −8.015105138192102002303708567947, −7.23133595769312761876945392762, −6.30732794327755624588542937743, −5.92107479072607268909853743268, −4.55784588257165667053773261181, −3.39853786914680186073742917184, −2.33727792920545830526310526613, −0.62791044201569185474029367226, 0.62791044201569185474029367226, 2.33727792920545830526310526613, 3.39853786914680186073742917184, 4.55784588257165667053773261181, 5.92107479072607268909853743268, 6.30732794327755624588542937743, 7.23133595769312761876945392762, 8.015105138192102002303708567947, 9.193634537030711191031976589522, 9.592472309668874924894894158066

Graph of the ZZ-function along the critical line