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Properties

Label 4-560e2-1.1-c1e2-0-29
Degree 44
Conductor 313600313600
Sign 11
Analytic cond. 19.995419.9954
Root an. cond. 2.114622.11462
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 5·9-s + 2·11-s − 8·19-s + 11·25-s + 2·29-s + 12·31-s − 20·41-s + 20·45-s − 49-s + 8·55-s + 12·59-s − 8·61-s + 32·71-s − 22·79-s + 16·81-s − 24·89-s − 32·95-s + 10·99-s + 30·109-s − 19·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯
L(s)  = 1  + 1.78·5-s + 5/3·9-s + 0.603·11-s − 1.83·19-s + 11/5·25-s + 0.371·29-s + 2.15·31-s − 3.12·41-s + 2.98·45-s − 1/7·49-s + 1.07·55-s + 1.56·59-s − 1.02·61-s + 3.79·71-s − 2.47·79-s + 16/9·81-s − 2.54·89-s − 3.28·95-s + 1.00·99-s + 2.87·109-s − 1.72·121-s + 2.14·125-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)=(313600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(313600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 313600313600    =    28⋅52⋅722^{8} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 19.995419.9954
Root analytic conductor: 2.114622.11462
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 313600, ( :1/2,1/2), 1)(4,\ 313600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1136807503.113680750
L(12)L(\frac12) ≈\approx 3.1136807503.113680750
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 1−4T+pT2 1 - 4 T + p T^{2}
7C2C_2 1+T2 1 + T^{2}
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
13C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
29C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
31C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
43C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
47C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
53C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
59C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
61C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
71C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
73C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
79C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
97C22C_2^2 1+167T2+p2T4 1 + 167 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

−10.70862297995796741768319571732, −10.32784410326647406069158074081, −9.992745442488012655709516755189, −9.954249134104325285537059379053, −9.380649780749393558087793542342, −8.737230499483633929453071530797, −8.488373296767513634131314970621, −8.029992865074042590792878729040, −7.11430035209296563389491696045, −6.77948403971659315174237405257, −6.40964565176020716426072231885, −6.27082841114926909906127331834, −5.30679125202027908688252562844, −5.01305810317297161823904958957, −4.35998212917175502672056735713, −3.98222919459956599204784297958, −3.06333061034806352466887240392, −2.29131801132705061393555228957, −1.77123885835003897391704340656, −1.16571607928687842361802773531, 1.16571607928687842361802773531, 1.77123885835003897391704340656, 2.29131801132705061393555228957, 3.06333061034806352466887240392, 3.98222919459956599204784297958, 4.35998212917175502672056735713, 5.01305810317297161823904958957, 5.30679125202027908688252562844, 6.27082841114926909906127331834, 6.40964565176020716426072231885, 6.77948403971659315174237405257, 7.11430035209296563389491696045, 8.029992865074042590792878729040, 8.488373296767513634131314970621, 8.737230499483633929453071530797, 9.380649780749393558087793542342, 9.954249134104325285537059379053, 9.992745442488012655709516755189, 10.32784410326647406069158074081, 10.70862297995796741768319571732

Graph of the ZZ-function along the critical line