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Properties

Label 4-2695275-1.1-c1e2-0-34
Degree 44
Conductor 26952752695275
Sign 11
Analytic cond. 171.853171.853
Root an. cond. 3.620673.62067
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  + 3·4-s + 4·5-s + 11-s + 5·16-s + 12·20-s + 11·25-s − 8·31-s + 3·44-s + 10·49-s + 4·55-s + 8·59-s + 3·64-s + 20·80-s + 33·100-s + 121-s − 24·124-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.78·5-s + 0.301·11-s + 5/4·16-s + 2.68·20-s + 11/5·25-s − 1.43·31-s + 0.452·44-s + 10/7·49-s + 0.539·55-s + 1.04·59-s + 3/8·64-s + 2.23·80-s + 3.29·100-s + 1/11·121-s − 2.15·124-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=(2695275s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2695275s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 26952752695275    =    34⋅52⋅1133^{4} \cdot 5^{2} \cdot 11^{3}
Sign: 11
Analytic conductor: 171.853171.853
Root analytic conductor: 3.620673.62067
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2695275, ( :1/2,1/2), 1)(4,\ 2695275,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.7333687765.733368776
L(12)L(\frac12) ≈\approx 5.7333687765.733368776
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)
bad3 1 1
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
11C1C_1 1−T 1 - T
good2C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−142T2+p2T4 1 - 142 T^{2} + p^{2} T^{4}
79C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
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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

−7.44071012856179454174115168498, −6.98461506866828385433924515992, −6.84884806265829454446157697534, −6.30325696612682555597887492081, −6.02499817336897508665224642231, −5.48878341540355437707060982436, −5.42457431972828552322534326719, −4.72801673816053907358368854207, −4.05560224434393851290900259231, −3.51309106430050388189908891790, −2.87537560535593228966647235605, −2.51187495471372643516646191681, −1.93873899637775283933678594487, −1.69569533896967354908280090098, −0.912271776203355030874824007294, 0.912271776203355030874824007294, 1.69569533896967354908280090098, 1.93873899637775283933678594487, 2.51187495471372643516646191681, 2.87537560535593228966647235605, 3.51309106430050388189908891790, 4.05560224434393851290900259231, 4.72801673816053907358368854207, 5.42457431972828552322534326719, 5.48878341540355437707060982436, 6.02499817336897508665224642231, 6.30325696612682555597887492081, 6.84884806265829454446157697534, 6.98461506866828385433924515992, 7.44071012856179454174115168498

Graph of the ZZ-function along the critical line