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Properties

Label 2-4650-1.1-c1-0-11
Degree 22
Conductor 46504650
Sign 11
Analytic cond. 37.130437.1304
Root an. cond. 6.093476.09347
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 − 2·7-s − 8-s + 9-s − 12-s + 4·13-s + 2·14-s + 16-s − 6·17-s − 18-s + 8·19-s + 2·21-s + 24-s − 4·26-s − 27-s − 2·28-s + 31-s − 32-s + 6·34-s + 36-s + 4·37-s − 8·38-s − 4·39-s − 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.436·21-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.377·28-s + 0.179·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.657·37-s − 1.29·38-s − 0.640·39-s − 0.937·41-s + ⋯

Functional equation

Λ(s)=(4650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46504650    =    2⋅3⋅52⋅312 \cdot 3 \cdot 5^{2} \cdot 31
Sign: 11
Analytic conductor: 37.130437.1304
Root analytic conductor: 6.093476.09347
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4650, ( :1/2), 1)(2,\ 4650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.90744823900.9074482390
L(12)L(\frac12) ≈\approx 0.90744823900.9074482390
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
5 1 1
31 1−T 1 - T
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+pT2 1 + 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

−8.460068752185368968389030422776, −7.47689438163177606510948862438, −6.89603122452971426718015346961, −6.22288671143227845702974054753, −5.62897566964802008811404209196, −4.63899227950574788419618160684, −3.64766166759127670855842491787, −2.86396218640878501834488499395, −1.64997389686777790527081002256, −0.61974766652524967827378665150, 0.61974766652524967827378665150, 1.64997389686777790527081002256, 2.86396218640878501834488499395, 3.64766166759127670855842491787, 4.63899227950574788419618160684, 5.62897566964802008811404209196, 6.22288671143227845702974054753, 6.89603122452971426718015346961, 7.47689438163177606510948862438, 8.460068752185368968389030422776

Graph of the ZZ-function along the critical line