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Properties

Label 4-395136-1.1-c1e2-0-35
Degree 44
Conductor 395136395136
Sign −1-1
Analytic cond. 25.194225.1942
Root an. cond. 2.240392.24039
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  + 3-s + 7-s − 2·9-s − 6·19-s + 21-s + 8·25-s − 5·27-s − 14·29-s − 5·31-s + 5·37-s + 2·47-s + 49-s − 4·53-s − 6·57-s − 59-s − 2·63-s + 8·75-s + 81-s − 2·83-s − 14·87-s − 5·93-s + 21·103-s − 20·109-s + 5·111-s + 113-s − 3·121-s + 127-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.37·19-s + 0.218·21-s + 8/5·25-s − 0.962·27-s − 2.59·29-s − 0.898·31-s + 0.821·37-s + 0.291·47-s + 1/7·49-s − 0.549·53-s − 0.794·57-s − 0.130·59-s − 0.251·63-s + 0.923·75-s + 1/9·81-s − 0.219·83-s − 1.50·87-s − 0.518·93-s + 2.06·103-s − 1.91·109-s + 0.474·111-s + 0.0940·113-s − 0.272·121-s + 0.0887·127-s + ⋯

Functional equation

Λ(s)=(395136s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(395136s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 395136395136    =    27⋅32⋅732^{7} \cdot 3^{2} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 25.194225.1942
Root analytic conductor: 2.240392.24039
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 395136, ( :1/2,1/2), −1)(4,\ 395136,\ (\ :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
3C2C_2 1−T+pT2 1 - T + p T^{2}
7C1C_1 1−T 1 - T
good5C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
11C22C_2^2 1+3T2+p2T4 1 + 3 T^{2} + p^{2} T^{4}
13C22C_2^2 1+9T2+p2T4 1 + 9 T^{2} + p^{2} T^{4}
17C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
19C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2×\timesC2C_2 (1+5T+pT2)(1+9T+pT2) ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−3T+pT2)(1+8T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−8T+pT2)(1+3T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} )
41C22C_2^2 1+31T2+p2T4 1 + 31 T^{2} + p^{2} T^{4}
43C22C_2^2 1−68T2+p2T4 1 - 68 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−3T+pT2)(1+7T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} )
59C2C_2×\timesC2C_2 (1−2T+pT2)(1+3T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} )
61C22C_2^2 1+29T2+p2T4 1 + 29 T^{2} + p^{2} T^{4}
67C22C_2^2 1+32T2+p2T4 1 + 32 T^{2} + p^{2} T^{4}
71C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
73C22C_2^2 1−88T2+p2T4 1 - 88 T^{2} + p^{2} T^{4}
79C22C_2^2 1+117T2+p2T4 1 + 117 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
89C22C_2^2 1+65T2+p2T4 1 + 65 T^{2} + p^{2} T^{4}
97C22C_2^2 1+36T2+p2T4 1 + 36 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

−8.542535040781039612494445646059, −7.898054090850495389633143027672, −7.64588185306719147061055876738, −7.19675747119123704152837612803, −6.48249917012102815535190335274, −6.18306273735141484451776377242, −5.47349157307834204550827486455, −5.19506344531750168930858623922, −4.48489371954854980466563637664, −3.88010194101076767996684853810, −3.49358032731442054253158539592, −2.67002041621380146562851251198, −2.22775970655050359414209924387, −1.45093646265834648153230174659, 0, 1.45093646265834648153230174659, 2.22775970655050359414209924387, 2.67002041621380146562851251198, 3.49358032731442054253158539592, 3.88010194101076767996684853810, 4.48489371954854980466563637664, 5.19506344531750168930858623922, 5.47349157307834204550827486455, 6.18306273735141484451776377242, 6.48249917012102815535190335274, 7.19675747119123704152837612803, 7.64588185306719147061055876738, 7.898054090850495389633143027672, 8.542535040781039612494445646059

Graph of the ZZ-function along the critical line