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Properties

Label 4-910e2-1.1-c1e2-0-17
Degree 44
Conductor 828100828100
Sign 11
Analytic cond. 52.800352.8003
Root an. cond. 2.695622.69562
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 − 4-s − 3·5-s + 4·7-s − 3·8-s − 3·10-s + 5·13-s + 4·14-s − 16-s + 3·20-s + 4·25-s + 5·26-s − 4·28-s − 10·29-s + 5·32-s − 12·35-s − 10·37-s + 9·40-s + 10·47-s + 9·49-s + 4·50-s − 5·52-s − 12·56-s − 10·58-s + 7·64-s − 15·65-s + 17·67-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.34·5-s + 1.51·7-s − 1.06·8-s − 0.948·10-s + 1.38·13-s + 1.06·14-s − 1/4·16-s + 0.670·20-s + 4/5·25-s + 0.980·26-s − 0.755·28-s − 1.85·29-s + 0.883·32-s − 2.02·35-s − 1.64·37-s + 1.42·40-s + 1.45·47-s + 9/7·49-s + 0.565·50-s − 0.693·52-s − 1.60·56-s − 1.31·58-s + 7/8·64-s − 1.86·65-s + 2.07·67-s + ⋯

Functional equation

Λ(s)=(828100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(828100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 828100828100    =    22⋅52⋅72⋅1322^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 52.800352.8003
Root analytic conductor: 2.695622.69562
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 828100, ( :1/2,1/2), 1)(4,\ 828100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8991838751.899183875
L(12)L(\frac12) ≈\approx 1.8991838751.899183875
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+pT2 1 - T + p T^{2}
5C2C_2 1+3T+pT2 1 + 3 T + p T^{2}
7C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
13C2C_2 1−5T+pT2 1 - 5 T + p T^{2}
good3C22C_2^2 1+p2T4 1 + p^{2} T^{4}
11C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
17C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
19C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
23C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1+pT2)(1+10T+pT2) ( 1 + p T^{2} )( 1 + 10 T + p T^{2} )
31C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1+pT2)(1+10T+pT2) ( 1 + p T^{2} )( 1 + 10 T + p T^{2} )
41C22C_2^2 1+27T2+p2T4 1 + 27 T^{2} + p^{2} T^{4}
43C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−8T+pT2)(1−2T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )
53C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
59C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−15T+pT2)(1−2T+pT2) ( 1 - 15 T + p T^{2} )( 1 - 2 T + p T^{2} )
71C22C_2^2 1+20T2+p2T4 1 + 20 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−2T+pT2)(1+12T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+3T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−6T+pT2)(1+11T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} )
89C22C_2^2 1−65T2+p2T4 1 - 65 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−19T+pT2)(1−18T+pT2) ( 1 - 19 T + p T^{2} )( 1 - 18 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

−8.384907606716604755146909068134, −7.72563785602142736028858164054, −7.45584937274136543385473227213, −7.03495957309274642467837366035, −6.31337650872544824735140754339, −5.75902371009033282955720019688, −5.48507105789302802646487835099, −4.85638567953984695031278672420, −4.56612656321565681987255903759, −3.89945272797370043233009908055, −3.70934164968343504770018970371, −3.30226944818049256621107695580, −2.28364480233662647720881213201, −1.56690003188077209026588135426, −0.62242832475543800281374199005, 0.62242832475543800281374199005, 1.56690003188077209026588135426, 2.28364480233662647720881213201, 3.30226944818049256621107695580, 3.70934164968343504770018970371, 3.89945272797370043233009908055, 4.56612656321565681987255903759, 4.85638567953984695031278672420, 5.48507105789302802646487835099, 5.75902371009033282955720019688, 6.31337650872544824735140754339, 7.03495957309274642467837366035, 7.45584937274136543385473227213, 7.72563785602142736028858164054, 8.384907606716604755146909068134

Graph of the ZZ-function along the critical line