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Properties

Label 4-704e2-1.1-c2e2-0-3
Degree 44
Conductor 495616495616
Sign 11
Analytic cond. 367.972367.972
Root an. cond. 4.379794.37979
Motivic weight 22
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  − 10·5-s + 7·9-s + 24·13-s + 28·17-s + 25·25-s + 48·29-s − 102·37-s + 80·41-s − 70·45-s + 54·49-s − 92·53-s + 88·61-s − 240·65-s + 88·73-s − 32·81-s − 280·85-s − 178·89-s + 370·97-s − 132·101-s + 364·109-s + 314·113-s + 168·117-s − 11·121-s + 250·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2·5-s + 7/9·9-s + 1.84·13-s + 1.64·17-s + 25-s + 1.65·29-s − 2.75·37-s + 1.95·41-s − 1.55·45-s + 1.10·49-s − 1.73·53-s + 1.44·61-s − 3.69·65-s + 1.20·73-s − 0.395·81-s − 3.29·85-s − 2·89-s + 3.81·97-s − 1.30·101-s + 3.33·109-s + 2.77·113-s + 1.43·117-s − 0.0909·121-s + 2·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

Λ(s)=(495616s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(495616s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 495616495616    =    212⋅1122^{12} \cdot 11^{2}
Sign: 11
Analytic conductor: 367.972367.972
Root analytic conductor: 4.379794.37979
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 495616, ( :1,1), 1)(4,\ 495616,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.0989785762.098978576
L(12)L(\frac12) ≈\approx 2.0989785762.098978576
L(2)L(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
11C2C_2 1+pT2 1 + p T^{2}
good3C2C_2 (1−5T+p2T2)(1+5T+p2T2) ( 1 - 5 T + p^{2} T^{2} )( 1 + 5 T + p^{2} T^{2} )
5C2C_2 (1+pT+p2T2)2 ( 1 + p T + p^{2} T^{2} )^{2}
7C22C_2^2 1−54T2+p4T4 1 - 54 T^{2} + p^{4} T^{4}
13C2C_2 (1−12T+p2T2)2 ( 1 - 12 T + p^{2} T^{2} )^{2}
17C2C_2 (1−14T+p2T2)2 ( 1 - 14 T + p^{2} T^{2} )^{2}
19C22C_2^2 1−18T2+p4T4 1 - 18 T^{2} + p^{4} T^{4}
23C22C_2^2 1−1047T2+p4T4 1 - 1047 T^{2} + p^{4} T^{4}
29C2C_2 (1−24T+p2T2)2 ( 1 - 24 T + p^{2} T^{2} )^{2}
31C22C_2^2 1−591T2+p4T4 1 - 591 T^{2} + p^{4} T^{4}
37C2C_2 (1+51T+p2T2)2 ( 1 + 51 T + p^{2} T^{2} )^{2}
41C2C_2 (1−40T+p2T2)2 ( 1 - 40 T + p^{2} T^{2} )^{2}
43C22C_2^2 1−134T2+p4T4 1 - 134 T^{2} + p^{4} T^{4}
47C22C_2^2 1+4206T2+p4T4 1 + 4206 T^{2} + p^{4} T^{4}
53C2C_2 (1+46T+p2T2)2 ( 1 + 46 T + p^{2} T^{2} )^{2}
59C22C_2^2 1−5631T2+p4T4 1 - 5631 T^{2} + p^{4} T^{4}
61C2C_2 (1−44T+p2T2)2 ( 1 - 44 T + p^{2} T^{2} )^{2}
67C22C_2^2 1−4127T2+p4T4 1 - 4127 T^{2} + p^{4} T^{4}
71C22C_2^2 1−6903T2+p4T4 1 - 6903 T^{2} + p^{4} T^{4}
73C2C_2 (1−44T+p2T2)2 ( 1 - 44 T + p^{2} T^{2} )^{2}
79C22C_2^2 1−12086T2+p4T4 1 - 12086 T^{2} + p^{4} T^{4}
83C22C_2^2 1+2106T2+p4T4 1 + 2106 T^{2} + p^{4} T^{4}
89C2C_2 (1+pT+p2T2)2 ( 1 + p T + p^{2} T^{2} )^{2}
97C2C_2 (1−185T+p2T2)2 ( 1 - 185 T + p^{2} T^{2} )^{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

−10.54094764262135898028814079720, −9.989946706439953340126475525675, −9.807089051928401843406539613529, −8.892381058138600834257333797296, −8.696032581055963454742862323237, −8.174679397039434925181927841747, −7.948321026391900012241550503650, −7.44324416771989257947224299819, −7.09183017479897049369334981027, −6.63033594887199001776033553500, −5.86578561939854134785483228241, −5.69660184191224083411928012484, −4.72215167866739939236819105737, −4.47376745861606238728966597585, −3.78886951960443329061591190523, −3.46599465655791031829510241732, −3.24193971221921742996838967101, −2.02514680299661315802694551036, −1.13120180409472766606793131913, −0.63304703990025272971206086134, 0.63304703990025272971206086134, 1.13120180409472766606793131913, 2.02514680299661315802694551036, 3.24193971221921742996838967101, 3.46599465655791031829510241732, 3.78886951960443329061591190523, 4.47376745861606238728966597585, 4.72215167866739939236819105737, 5.69660184191224083411928012484, 5.86578561939854134785483228241, 6.63033594887199001776033553500, 7.09183017479897049369334981027, 7.44324416771989257947224299819, 7.948321026391900012241550503650, 8.174679397039434925181927841747, 8.696032581055963454742862323237, 8.892381058138600834257333797296, 9.807089051928401843406539613529, 9.989946706439953340126475525675, 10.54094764262135898028814079720

Graph of the ZZ-function along the critical line