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Properties

Label 4-259200-1.1-c1e2-0-5
Degree 44
Conductor 259200259200
Sign 11
Analytic cond. 16.526816.5268
Root an. cond. 2.016262.01626
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  − 4·13-s + 8·23-s − 25-s + 4·37-s − 8·47-s + 6·49-s + 4·61-s + 16·71-s − 4·73-s + 16·83-s − 4·97-s + 16·107-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.10·13-s + 1.66·23-s − 1/5·25-s + 0.657·37-s − 1.16·47-s + 6/7·49-s + 0.512·61-s + 1.89·71-s − 0.468·73-s + 1.75·83-s − 0.406·97-s + 1.54·107-s − 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=(259200s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(259200s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 259200259200    =    27⋅34⋅522^{7} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 16.526816.5268
Root analytic conductor: 2.016262.01626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 259200, ( :1/2,1/2), 1)(4,\ 259200,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5665989331.566598933
L(12)L(\frac12) ≈\approx 1.5665989331.566598933
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
3 1 1
5C2C_2 1+T2 1 + T^{2}
good7C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{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+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
29C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
43C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1−14T+pT2)(1−2T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} )
89C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 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

−9.129296133530559885394075950226, −8.396523644088606422215069847706, −7.967211502850339151955928539718, −7.52839657271779380918414162346, −7.00137644920259989753610340300, −6.65023927439790795985709979733, −6.10280451823324955313981595609, −5.30061725162678769238788205992, −5.12147080240708820645410571852, −4.50651416000898318828262604108, −3.87811169898465142448625710841, −3.15822749708855078037531916407, −2.61762806168029183637732500482, −1.88500540257971488927426795945, −0.76158493647655685835419264301, 0.76158493647655685835419264301, 1.88500540257971488927426795945, 2.61762806168029183637732500482, 3.15822749708855078037531916407, 3.87811169898465142448625710841, 4.50651416000898318828262604108, 5.12147080240708820645410571852, 5.30061725162678769238788205992, 6.10280451823324955313981595609, 6.65023927439790795985709979733, 7.00137644920259989753610340300, 7.52839657271779380918414162346, 7.967211502850339151955928539718, 8.396523644088606422215069847706, 9.129296133530559885394075950226

Graph of the ZZ-function along the critical line