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Properties

Label 4-429e2-1.1-c1e2-0-1
Degree 44
Conductor 184041184041
Sign 11
Analytic cond. 11.734611.7346
Root an. cond. 1.850831.85083
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  + 4-s + 9-s − 2·13-s − 3·16-s + 2·25-s + 8·29-s + 36-s + 4·43-s + 10·49-s − 2·52-s + 12·53-s + 4·61-s − 7·64-s + 4·79-s + 81-s + 2·100-s − 24·101-s + 12·113-s + 8·116-s − 2·117-s + 121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s − 0.554·13-s − 3/4·16-s + 2/5·25-s + 1.48·29-s + 1/6·36-s + 0.609·43-s + 10/7·49-s − 0.277·52-s + 1.64·53-s + 0.512·61-s − 7/8·64-s + 0.450·79-s + 1/9·81-s + 1/5·100-s − 2.38·101-s + 1.12·113-s + 0.742·116-s − 0.184·117-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(184041s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(184041s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 184041184041    =    32⋅112⋅1323^{2} \cdot 11^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 11.734611.7346
Root analytic conductor: 1.850831.85083
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 184041, ( :1/2,1/2), 1)(4,\ 184041,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8546400351.854640035
L(12)L(\frac12) ≈\approx 1.8546400351.854640035
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)
bad3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
11C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
13C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good2C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
37C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
41C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−8T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
67C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
71C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−12T+pT2)(1+8T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
89C22C_2^2 1−138T2+p2T4 1 - 138 T^{2} + p^{2} T^{4}
97C22C_2^2 1+14T2+p2T4 1 + 14 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

−9.034675924042442143095749364854, −8.751255841935656095454046788480, −8.201720666313246234236012947159, −7.64207526479637796430779583830, −7.06106563280908413208522152193, −6.89083570242035191053138253645, −6.32264419475329643065103807654, −5.66441761296988050389009552569, −5.19940310794129448841020654032, −4.43271259079955941847260762295, −4.19246927836238336661479687820, −3.23556710348062248878122127648, −2.58345108108264936945916302095, −2.06329049460713332419606765221, −0.899003034750625609668785182244, 0.899003034750625609668785182244, 2.06329049460713332419606765221, 2.58345108108264936945916302095, 3.23556710348062248878122127648, 4.19246927836238336661479687820, 4.43271259079955941847260762295, 5.19940310794129448841020654032, 5.66441761296988050389009552569, 6.32264419475329643065103807654, 6.89083570242035191053138253645, 7.06106563280908413208522152193, 7.64207526479637796430779583830, 8.201720666313246234236012947159, 8.751255841935656095454046788480, 9.034675924042442143095749364854

Graph of the ZZ-function along the critical line