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Properties

Label 4-48e2-1.1-c1e2-0-2
Degree 44
Conductor 23042304
Sign 11
Analytic cond. 0.1469050.146905
Root an. cond. 0.6190970.619097
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  − 3·9-s − 4·13-s + 10·25-s − 20·37-s + 2·49-s + 28·61-s + 20·73-s + 9·81-s − 28·97-s − 4·109-s + 12·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 9-s − 1.10·13-s + 2·25-s − 3.28·37-s + 2/7·49-s + 3.58·61-s + 2.34·73-s + 81-s − 2.84·97-s − 0.383·109-s + 1.10·117-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

Λ(s)=(2304s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2304s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 23042304    =    28⋅322^{8} \cdot 3^{2}
Sign: 11
Analytic conductor: 0.1469050.146905
Root analytic conductor: 0.6190970.619097
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2304, ( :1/2,1/2), 1)(4,\ 2304,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.63851446470.6385144647
L(12)L(\frac12) ≈\approx 0.63851446470.6385144647
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)Isogeny Class over Fp\mathbf{F}_p
bad2 1 1
3C2C_2 1+pT2 1 + p T^{2}
good5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2} 2.5.a_ak
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) 2.7.a_ac
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.11.a_w
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2} 2.13.e_be
17C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2} 2.17.a_abi
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) 2.19.a_aba
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.23.a_bu
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2} 2.29.a_acg
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) 2.31.a_bu
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2} 2.37.u_gs
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2} 2.41.a_ade
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) 2.43.a_w
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.47.a_dq
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2} 2.53.a_aec
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.59.a_eo
61C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2} 2.61.abc_mg
67C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) 2.67.a_aes
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.71.a_fm
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2} 2.73.au_jm
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) 2.79.a_fm
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.83.a_gk
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2} 2.89.a_agw
97C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2} 2.97.bc_pa
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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

−15.92369172058273067985032747780, −15.44375138708661754247537486481, −14.60504122510036768778392573017, −14.46956971764483229019315726323, −13.83864131223898957951710912710, −13.12925026034704855482017874281, −12.24056695896816489745402282108, −12.23659147913522161754220310176, −11.26412486568634322020379293105, −10.73331120502494611207295480445, −10.10407073322686381731412723868, −9.338892356501978764549936387737, −8.625402204712350869792689549676, −8.201099013122993350293745119564, −7.02451456717377461766522669630, −6.74418422965018797428766625394, −5.37448383935061293331068662521, −5.06935218801281721891351547462, −3.63537616080026223751227237238, −2.52494733590377164458054694339, 2.52494733590377164458054694339, 3.63537616080026223751227237238, 5.06935218801281721891351547462, 5.37448383935061293331068662521, 6.74418422965018797428766625394, 7.02451456717377461766522669630, 8.201099013122993350293745119564, 8.625402204712350869792689549676, 9.338892356501978764549936387737, 10.10407073322686381731412723868, 10.73331120502494611207295480445, 11.26412486568634322020379293105, 12.23659147913522161754220310176, 12.24056695896816489745402282108, 13.12925026034704855482017874281, 13.83864131223898957951710912710, 14.46956971764483229019315726323, 14.60504122510036768778392573017, 15.44375138708661754247537486481, 15.92369172058273067985032747780

Graph of the ZZ-function along the critical line