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Properties

Label 4-46208-1.1-c1e2-0-1
Degree 44
Conductor 4620846208
Sign 11
Analytic cond. 2.946262.94626
Root an. cond. 1.310141.31014
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  + 2-s + 4-s + 6·7-s + 8-s − 5·9-s + 6·14-s + 16-s + 6·17-s − 5·18-s − 2·23-s + 6·25-s + 6·28-s − 16·31-s + 32-s + 6·34-s − 5·36-s − 16·41-s − 2·46-s + 16·47-s + 13·49-s + 6·50-s + 6·56-s − 16·62-s − 30·63-s + 64-s + 6·68-s + 4·71-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 2.26·7-s + 0.353·8-s − 5/3·9-s + 1.60·14-s + 1/4·16-s + 1.45·17-s − 1.17·18-s − 0.417·23-s + 6/5·25-s + 1.13·28-s − 2.87·31-s + 0.176·32-s + 1.02·34-s − 5/6·36-s − 2.49·41-s − 0.294·46-s + 2.33·47-s + 13/7·49-s + 0.848·50-s + 0.801·56-s − 2.03·62-s − 3.77·63-s + 1/8·64-s + 0.727·68-s + 0.474·71-s + ⋯

Functional equation

Λ(s)=(46208s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(46208s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 46208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4620846208    =    27⋅1922^{7} \cdot 19^{2}
Sign: 11
Analytic conductor: 2.946262.94626
Root analytic conductor: 1.310141.31014
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 46208, ( :1/2,1/2), 1)(4,\ 46208,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3729209822.372920982
L(12)L(\frac12) ≈\approx 2.3729209822.372920982
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)
bad2C1C_1 1−T 1 - T
19C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C2C_2 (1−3T+pT2)2 ( 1 - 3 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−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
23C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
29C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
59C2C_2 (1−15T+pT2)(1+15T+pT2) ( 1 - 15 T + p T^{2} )( 1 + 15 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 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
71C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
73C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
79C2C_2 (1+10T+pT2)2 ( 1 + 10 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+pT2)2 ( 1 + 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

−10.44209956582501917351653608378, −9.673084911618933118181680374348, −8.905991841422638352773015657402, −8.408486909011940238981299566497, −8.264845552131759886230905508728, −7.41068992206199799917476516665, −7.25038064772684706053554512263, −6.17646264461499634926610391599, −5.50702009116960948045704530045, −5.28069074518443602737935578596, −4.86338510531971803230901631456, −3.86057015395193111701613139713, −3.27846026315236304587310905813, −2.31224961625843070084602752549, −1.49207601742473373563863301755, 1.49207601742473373563863301755, 2.31224961625843070084602752549, 3.27846026315236304587310905813, 3.86057015395193111701613139713, 4.86338510531971803230901631456, 5.28069074518443602737935578596, 5.50702009116960948045704530045, 6.17646264461499634926610391599, 7.25038064772684706053554512263, 7.41068992206199799917476516665, 8.264845552131759886230905508728, 8.408486909011940238981299566497, 8.905991841422638352773015657402, 9.673084911618933118181680374348, 10.44209956582501917351653608378

Graph of the ZZ-function along the critical line