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Properties

Label 4-525e2-1.1-c1e2-0-5
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 17.574017.5740
Root an. cond. 2.047472.04747
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·7-s − 3·9-s − 3·16-s + 12·17-s − 4·28-s − 3·36-s + 4·37-s − 12·41-s + 16·43-s − 24·47-s + 9·49-s + 24·59-s + 12·63-s − 7·64-s − 16·67-s + 12·68-s + 16·79-s + 9·81-s − 12·89-s + 12·101-s − 4·109-s + 12·112-s − 48·119-s + 10·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.51·7-s − 9-s − 3/4·16-s + 2.91·17-s − 0.755·28-s − 1/2·36-s + 0.657·37-s − 1.87·41-s + 2.43·43-s − 3.50·47-s + 9/7·49-s + 3.12·59-s + 1.51·63-s − 7/8·64-s − 1.95·67-s + 1.45·68-s + 1.80·79-s + 81-s − 1.27·89-s + 1.19·101-s − 0.383·109-s + 1.13·112-s − 4.40·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 17.574017.5740
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 275625, ( :1/2,1/2), 1)(4,\ 275625,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3615202031.361520203
L(12)L(\frac12) ≈\approx 1.3615202031.361520203
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)
bad3C2C_2 1+pT2 1 + p T^{2}
5 1 1
7C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good2C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
67C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
71C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
73C22C_2^2 1−98T2+p2T4 1 - 98 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C22C_2^2 1−146T2+p2T4 1 - 146 T^{2} + 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

−11.45237090828949478474383705406, −10.47212270757445484131359350354, −10.04587377959947312920249244919, −9.977409902473546394359670122745, −9.225366481388965813566345209463, −9.133659500153687179359894363497, −8.162909626428848534701027055385, −8.111675385615524935608401307889, −7.47384913185477231515421335849, −6.76658042870903684333855826712, −6.69271271891388967550066955704, −5.87330229878917318829235879409, −5.71045363267218455269869724935, −5.17036163867387101142343237463, −4.36762313245250054001214211565, −3.46471743218789052078383852691, −3.25680015186414588701100659352, −2.79266163078902945878075225686, −1.88052529093017967831170943536, −0.68009710406838669263553681590, 0.68009710406838669263553681590, 1.88052529093017967831170943536, 2.79266163078902945878075225686, 3.25680015186414588701100659352, 3.46471743218789052078383852691, 4.36762313245250054001214211565, 5.17036163867387101142343237463, 5.71045363267218455269869724935, 5.87330229878917318829235879409, 6.69271271891388967550066955704, 6.76658042870903684333855826712, 7.47384913185477231515421335849, 8.111675385615524935608401307889, 8.162909626428848534701027055385, 9.133659500153687179359894363497, 9.225366481388965813566345209463, 9.977409902473546394359670122745, 10.04587377959947312920249244919, 10.47212270757445484131359350354, 11.45237090828949478474383705406

Graph of the ZZ-function along the critical line