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Properties

Label 4-910e2-1.1-c1e2-0-1
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 no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 5-s − 7-s − 4·8-s + 3·9-s + 2·10-s − 11-s − 5·13-s + 2·14-s + 5·16-s − 8·17-s − 6·18-s − 8·19-s − 3·20-s + 2·22-s + 10·23-s + 10·26-s − 3·28-s + 2·29-s − 10·31-s − 6·32-s + 16·34-s + 35-s + 9·36-s − 4·37-s + 16·38-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.447·5-s − 0.377·7-s − 1.41·8-s + 9-s + 0.632·10-s − 0.301·11-s − 1.38·13-s + 0.534·14-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 1.83·19-s − 0.670·20-s + 0.426·22-s + 2.08·23-s + 1.96·26-s − 0.566·28-s + 0.371·29-s − 1.79·31-s − 1.06·32-s + 2.74·34-s + 0.169·35-s + 3/2·36-s − 0.657·37-s + 2.59·38-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: no
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 0.17265773760.1726577376
L(12)L(\frac12) ≈\approx 0.17265773760.1726577376
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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
5C2C_2 1+T+T2 1 + T + T^{2}
7C2C_2 1+T+pT2 1 + T + p T^{2}
13C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good3C2C_2 (1−pT+pT2)(1+pT+pT2) ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )
11C22C_2^2 1+T−10T2+pT3+p2T4 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4}
17C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
19C2C_2 (1+T+pT2)(1+7T+pT2) ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} )
23C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
29C22C_2^2 1−2T−25T2−2pT3+p2T4 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+10T+69T2+10pT3+p2T4 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4}
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C22C_2^2 1+3T−32T2+3pT3+p2T4 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+6T−7T2+6pT3+p2T4 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+8T+17T2+8pT3+p2T4 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+5T−28T2+5pT3+p2T4 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C22C_2^2 1−8T+3T2−8pT3+p2T4 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+2T−63T2+2pT3+p2T4 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+12T+73T2+12pT3+p2T4 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+16T+183T2+16pT3+p2T4 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−2T−75T2−2pT3+p2T4 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4}
83C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
89C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
97C22C_2^2 1+8T−33T2+8pT3+p2T4 1 + 8 T - 33 T^{2} + 8 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

−10.52796594695979878199182042631, −9.795950206284353913610816065733, −9.459468581265115748865655139313, −8.991866193509581368405682127914, −8.795935821127456575658625617291, −8.134446316029763702820524619344, −8.043989383862278072754168478459, −7.11109157167891147442845683911, −7.05627752997572032139066521326, −6.61300089150476102490955354780, −6.60123053386520552157316841037, −5.32919773760748920058169549670, −5.22845520551690744136524376096, −4.37300743878545598567081083930, −4.14111075344903286009663480685, −3.24893740290304536718548029527, −2.67788104112472863392467872818, −2.07001723757237341404615016693, −1.57504599189052410109176622911, −0.23932874241198483053841117808, 0.23932874241198483053841117808, 1.57504599189052410109176622911, 2.07001723757237341404615016693, 2.67788104112472863392467872818, 3.24893740290304536718548029527, 4.14111075344903286009663480685, 4.37300743878545598567081083930, 5.22845520551690744136524376096, 5.32919773760748920058169549670, 6.60123053386520552157316841037, 6.61300089150476102490955354780, 7.05627752997572032139066521326, 7.11109157167891147442845683911, 8.043989383862278072754168478459, 8.134446316029763702820524619344, 8.795935821127456575658625617291, 8.991866193509581368405682127914, 9.459468581265115748865655139313, 9.795950206284353913610816065733, 10.52796594695979878199182042631

Graph of the ZZ-function along the critical line