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Properties

Label 4-170e2-1.1-c1e2-0-16
Degree 44
Conductor 2890028900
Sign 11
Analytic cond. 1.842681.84268
Root an. cond. 1.165091.16509
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 3·5-s + 2·6-s − 4·7-s + 8-s − 3·9-s + 3·10-s − 3·11-s + 4·14-s + 6·15-s − 16-s − 6·17-s + 3·18-s − 5·19-s + 8·21-s + 3·22-s − 2·24-s + 4·25-s + 14·27-s + 3·29-s − 6·30-s − 9·31-s + 6·33-s + 6·34-s + 12·35-s + 8·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1.34·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s − 9-s + 0.948·10-s − 0.904·11-s + 1.06·14-s + 1.54·15-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 1.14·19-s + 1.74·21-s + 0.639·22-s − 0.408·24-s + 4/5·25-s + 2.69·27-s + 0.557·29-s − 1.09·30-s − 1.61·31-s + 1.04·33-s + 1.02·34-s + 2.02·35-s + 1.31·37-s + ⋯

Functional equation

Λ(s)=(28900s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(28900s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2890028900    =    22⋅52⋅1722^{2} \cdot 5^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 1.842681.84268
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 28900, ( :1/2,1/2), 1)(4,\ 28900,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1+T+T2 1 + T + T^{2}
5C2C_2 1+3T+pT2 1 + 3 T + p T^{2}
17C2C_2 1+6T+pT2 1 + 6 T + p T^{2}
good3C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
7C2C_2 (1−T+pT2)(1+5T+pT2) ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} )
11D4D_{4} 1+3T+7T2+3pT3+p2T4 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
19D4D_{4} 1+5T+33T2+5pT3+p2T4 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4}
23C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
29D4D_{4} 1−3T−2T2−3pT3+p2T4 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+9T+58T2+9pT3+p2T4 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
41D4D_{4} 1+9T+67T2+9pT3+p2T4 1 + 9 T + 67 T^{2} + 9 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−9T+61T2−9pT3+p2T4 1 - 9 T + 61 T^{2} - 9 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+6T+7T2+6pT3+p2T4 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−44T2+p2T4 1 - 44 T^{2} + p^{2} T^{4}
59D4D_{4} 1−3T+19T2−3pT3+p2T4 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+6T+pT2+6pT3+p2T4 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+15T+121T2+15pT3+p2T4 1 + 15 T + 121 T^{2} + 15 p T^{3} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1+pT2)(1+15T+pT2) ( 1 + p T^{2} )( 1 + 15 T + p T^{2} )
73D4D_{4} 1−2T−60T2−2pT3+p2T4 1 - 2 T - 60 T^{2} - 2 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+67T2+p2T4 1 + 67 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1−9T+pT2)(1+12T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} )
89D4D_{4} 1+15T+196T2+15pT3+p2T4 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+T−105T2+pT3+p2T4 1 + T - 105 T^{2} + p T^{3} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−16.0221711185, −15.5342970694, −14.9897584332, −14.6635859038, −13.8099113219, −13.2907075077, −12.8523853416, −12.4418385025, −11.9521338934, −11.3387256738, −11.0645280668, −10.6807001131, −10.1971453750, −9.45577863031, −8.77312346071, −8.61883253709, −7.95635897687, −7.25503634354, −6.73099839433, −6.12036715031, −5.72019169432, −4.78117523089, −4.24511295137, −3.26424045735, −2.58974577135, 0, 0, 2.58974577135, 3.26424045735, 4.24511295137, 4.78117523089, 5.72019169432, 6.12036715031, 6.73099839433, 7.25503634354, 7.95635897687, 8.61883253709, 8.77312346071, 9.45577863031, 10.1971453750, 10.6807001131, 11.0645280668, 11.3387256738, 11.9521338934, 12.4418385025, 12.8523853416, 13.2907075077, 13.8099113219, 14.6635859038, 14.9897584332, 15.5342970694, 16.0221711185

Graph of the ZZ-function along the critical line