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Properties

Label 4-2e14-1.1-c1e2-0-4
Degree 44
Conductor 1638416384
Sign 11
Analytic cond. 1.044651.04465
Root an. cond. 1.010981.01098
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·7-s − 2·9-s − 4·17-s − 8·23-s − 6·25-s − 12·41-s + 16·47-s + 34·49-s − 16·63-s − 24·71-s + 28·73-s + 16·79-s − 5·81-s − 4·89-s − 4·97-s + 8·103-s + 4·113-s − 32·119-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + ⋯
L(s)  = 1  + 3.02·7-s − 2/3·9-s − 0.970·17-s − 1.66·23-s − 6/5·25-s − 1.87·41-s + 2.33·47-s + 34/7·49-s − 2.01·63-s − 2.84·71-s + 3.27·73-s + 1.80·79-s − 5/9·81-s − 0.423·89-s − 0.406·97-s + 0.788·103-s + 0.376·113-s − 2.93·119-s − 1.63·121-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 + 0.646·153-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=(16384s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(16384s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1638416384    =    2142^{14}
Sign: 11
Analytic conductor: 1.044651.04465
Root analytic conductor: 1.010981.01098
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 16384, ( :1/2,1/2), 1)(4,\ 16384,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3343021111.334302111
L(12)L(\frac12) ≈\approx 1.3343021111.334302111
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
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1+2T+pT2)2 ( 1 + 2 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

−11.28853583105345723449818824184, −10.57265847830629711938727340297, −10.22941768705266383849925633731, −9.215939702261382322141825805489, −8.728992519223720851294830795286, −8.098681382024412731658064042949, −7.984207807149970007419508597610, −7.33738117792255002879146989083, −6.39871498618546548325782647590, −5.59100833138318705450052051530, −5.16542238567969836254694300670, −4.43324404972048918127214240916, −3.91058908136761919842529082491, −2.30676446591740616355718333113, −1.73703685485230483844290438322, 1.73703685485230483844290438322, 2.30676446591740616355718333113, 3.91058908136761919842529082491, 4.43324404972048918127214240916, 5.16542238567969836254694300670, 5.59100833138318705450052051530, 6.39871498618546548325782647590, 7.33738117792255002879146989083, 7.984207807149970007419508597610, 8.098681382024412731658064042949, 8.728992519223720851294830795286, 9.215939702261382322141825805489, 10.22941768705266383849925633731, 10.57265847830629711938727340297, 11.28853583105345723449818824184

Graph of the ZZ-function along the critical line