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Properties

Label 4-48e4-1.1-c1e2-0-6
Degree 44
Conductor 53084165308416
Sign 11
Analytic cond. 338.469338.469
Root an. cond. 4.289234.28923
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  − 12·17-s + 8·19-s + 2·25-s − 12·41-s + 8·43-s − 2·49-s + 24·59-s − 8·67-s − 4·73-s + 12·89-s − 4·97-s + 24·107-s + 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.91·17-s + 1.83·19-s + 2/5·25-s − 1.87·41-s + 1.21·43-s − 2/7·49-s + 3.12·59-s − 0.977·67-s − 0.468·73-s + 1.27·89-s − 0.406·97-s + 2.32·107-s + 1.12·113-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 − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=(5308416s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5308416s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 53084165308416    =    216⋅342^{16} \cdot 3^{4}
Sign: 11
Analytic conductor: 338.469338.469
Root analytic conductor: 4.289234.28923
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 5308416, ( :1/2,1/2), 1)(4,\ 5308416,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7185603671.718560367
L(12)L(\frac12) ≈\approx 1.7185603671.718560367
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
good5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
31C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
37C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
53C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p 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

−9.051958995200110935297158929877, −8.872238150144058509099300033791, −8.393135543397082230969700621486, −8.297128464064606045304138325176, −7.40784544888046377444052101226, −7.33661936490695083674831742076, −6.87296624703370658242518001585, −6.63533842994695874337372444054, −6.00748604342774431415303044952, −5.78738706852668711508938914379, −5.05640537320441894785617631505, −4.89148071841182910687091315400, −4.47073029822226100270455099896, −3.88816374686518382818613341114, −3.50402801172082537256356145911, −2.95041560502240512540940991021, −2.34522358440884180999113886745, −2.04697187072846738510564596377, −1.24474527390506146238494436048, −0.47076910174565786246861024592, 0.47076910174565786246861024592, 1.24474527390506146238494436048, 2.04697187072846738510564596377, 2.34522358440884180999113886745, 2.95041560502240512540940991021, 3.50402801172082537256356145911, 3.88816374686518382818613341114, 4.47073029822226100270455099896, 4.89148071841182910687091315400, 5.05640537320441894785617631505, 5.78738706852668711508938914379, 6.00748604342774431415303044952, 6.63533842994695874337372444054, 6.87296624703370658242518001585, 7.33661936490695083674831742076, 7.40784544888046377444052101226, 8.297128464064606045304138325176, 8.393135543397082230969700621486, 8.872238150144058509099300033791, 9.051958995200110935297158929877

Graph of the ZZ-function along the critical line