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Properties

Label 4-2695275-1.1-c1e2-0-19
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 − 2·5-s + 11-s + 5·16-s − 6·20-s − 25-s + 16·31-s + 3·44-s − 2·49-s − 2·55-s + 8·59-s + 3·64-s − 10·80-s − 12·89-s − 3·100-s + 121-s + 48·124-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + ⋯
L(s)  = 1  + 3/2·4-s − 0.894·5-s + 0.301·11-s + 5/4·16-s − 1.34·20-s − 1/5·25-s + 2.87·31-s + 0.452·44-s − 2/7·49-s − 0.269·55-s + 1.04·59-s + 3/8·64-s − 1.11·80-s − 1.27·89-s − 0.299·100-s + 1/11·121-s + 4.31·124-s + 1.07·125-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 + ⋯

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 2.9501863452.950186345
L(12)L(\frac12) ≈\approx 2.9501863452.950186345
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+2T+pT2 1 + 2 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+2T2+p2T4 1 + 2 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+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 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 (1−8T+pT2)2 ( 1 - 8 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 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{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 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + 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

−7.60290232847304408135830133156, −7.06357245241558270656855619159, −6.86156063367989210039158772723, −6.46286624390891316695752121741, −6.02045916898048232106941994126, −5.70161555368798107683496311288, −4.97199342804118720933760459970, −4.58491876123190085009592262009, −4.04083673948420490884258308720, −3.64920117372470453410488274693, −2.90668071449851211497557084111, −2.76291904484387861999019027273, −2.06493997072077583121716954000, −1.41914140794236273960317365267, −0.67051293034997416873087953327, 0.67051293034997416873087953327, 1.41914140794236273960317365267, 2.06493997072077583121716954000, 2.76291904484387861999019027273, 2.90668071449851211497557084111, 3.64920117372470453410488274693, 4.04083673948420490884258308720, 4.58491876123190085009592262009, 4.97199342804118720933760459970, 5.70161555368798107683496311288, 6.02045916898048232106941994126, 6.46286624390891316695752121741, 6.86156063367989210039158772723, 7.06357245241558270656855619159, 7.60290232847304408135830133156

Graph of the ZZ-function along the critical line