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Properties

Label 4-48e3-1.1-c1e2-0-3
Degree 44
Conductor 110592110592
Sign 11
Analytic cond. 7.051447.05144
Root an. cond. 1.629551.62955
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  + 3-s + 9-s − 4·13-s + 8·23-s + 2·25-s + 27-s + 4·37-s − 4·39-s + 8·47-s + 2·49-s + 24·59-s + 4·61-s + 8·69-s + 8·71-s − 12·73-s + 2·75-s + 81-s − 16·83-s − 12·97-s + 8·107-s + 12·109-s + 4·111-s − 4·117-s − 6·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.10·13-s + 1.66·23-s + 2/5·25-s + 0.192·27-s + 0.657·37-s − 0.640·39-s + 1.16·47-s + 2/7·49-s + 3.12·59-s + 0.512·61-s + 0.963·69-s + 0.949·71-s − 1.40·73-s + 0.230·75-s + 1/9·81-s − 1.75·83-s − 1.21·97-s + 0.773·107-s + 1.14·109-s + 0.379·111-s − 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=(110592s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(110592s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 110592110592    =    212⋅332^{12} \cdot 3^{3}
Sign: 11
Analytic conductor: 7.051447.05144
Root analytic conductor: 1.629551.62955
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 110592, ( :1/2,1/2), 1)(4,\ 110592,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8828877301.882887730
L(12)L(\frac12) ≈\approx 1.8828877301.882887730
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
3C1C_1 1−T 1 - T
good5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
19C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
29C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
31C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
43C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
53C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
73C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1+4T+pT2)(1+12T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
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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

−9.441458527347827045314306362484, −9.031290177123584089968069673475, −8.505176949138939806765746980171, −8.173514554803442734671541133944, −7.32328213459775648078965393123, −7.15170496964375224529488571068, −6.75011296795482591362571637167, −5.82998430379572433726607082968, −5.36491761982195510321568888557, −4.75117524706476146238747265245, −4.22069817552665673087881290479, −3.47738164570821613705383822117, −2.73018428216902765717531355398, −2.27617566150396406286489395179, −1.01010663988588321749755795699, 1.01010663988588321749755795699, 2.27617566150396406286489395179, 2.73018428216902765717531355398, 3.47738164570821613705383822117, 4.22069817552665673087881290479, 4.75117524706476146238747265245, 5.36491761982195510321568888557, 5.82998430379572433726607082968, 6.75011296795482591362571637167, 7.15170496964375224529488571068, 7.32328213459775648078965393123, 8.173514554803442734671541133944, 8.505176949138939806765746980171, 9.031290177123584089968069673475, 9.441458527347827045314306362484

Graph of the ZZ-function along the critical line