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Properties

Label 4-320e2-1.1-c1e2-0-45
Degree 44
Conductor 102400102400
Sign −1-1
Analytic cond. 6.529116.52911
Root an. cond. 1.598501.59850
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·9-s − 4·11-s − 12·19-s − 25-s − 4·29-s − 4·41-s − 4·45-s + 6·49-s − 8·55-s − 4·59-s − 12·61-s − 8·71-s + 16·79-s − 5·81-s + 4·89-s − 24·95-s + 8·99-s + 12·101-s + 20·109-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s − 1.20·11-s − 2.75·19-s − 1/5·25-s − 0.742·29-s − 0.624·41-s − 0.596·45-s + 6/7·49-s − 1.07·55-s − 0.520·59-s − 1.53·61-s − 0.949·71-s + 1.80·79-s − 5/9·81-s + 0.423·89-s − 2.46·95-s + 0.804·99-s + 1.19·101-s + 1.91·109-s − 0.545·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + ⋯

Functional equation

Λ(s)=(102400s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(102400s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 102400102400    =    212⋅522^{12} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 6.529116.52911
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 102400, ( :1/2,1/2), −1)(4,\ 102400,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
17C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1+4T+pT2)(1+8T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
47C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
53C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−8T+pT2)(1+12T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
67C22C_2^2 1+66T2+p2T4 1 + 66 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
73C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−62T2+p2T4 1 - 62 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C22C_2^2 1−18T2+p2T4 1 - 18 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

−9.053741841487707353882126684056, −8.990015121740525383648379496576, −8.373164867544173382186980638339, −7.87156441916130722244009057393, −7.41031075946135687926852450580, −6.61127426785943061529500519900, −6.15692190774184236929415211667, −5.83117384836379108269028934953, −5.18695218824486532825846447060, −4.62582934312640197935671693414, −3.95524337065984516997232290540, −3.07982062010987887626317506271, −2.32406872197087040622380442993, −1.89724448843746106184694344043, 0, 1.89724448843746106184694344043, 2.32406872197087040622380442993, 3.07982062010987887626317506271, 3.95524337065984516997232290540, 4.62582934312640197935671693414, 5.18695218824486532825846447060, 5.83117384836379108269028934953, 6.15692190774184236929415211667, 6.61127426785943061529500519900, 7.41031075946135687926852450580, 7.87156441916130722244009057393, 8.373164867544173382186980638339, 8.990015121740525383648379496576, 9.053741841487707353882126684056

Graph of the ZZ-function along the critical line