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Properties

Label 4-43904-1.1-c1e2-0-15
Degree 44
Conductor 4390443904
Sign −1-1
Analytic cond. 2.799352.79935
Root an. cond. 1.293491.29349
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 2·9-s − 8·13-s − 14-s + 16-s + 2·18-s − 10·25-s + 8·26-s + 28-s − 8·31-s − 32-s − 2·36-s + 16·43-s − 24·47-s + 49-s + 10·50-s − 8·52-s − 56-s + 16·61-s + 8·62-s − 2·63-s + 64-s − 8·67-s + 2·72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 2.21·13-s − 0.267·14-s + 1/4·16-s + 0.471·18-s − 2·25-s + 1.56·26-s + 0.188·28-s − 1.43·31-s − 0.176·32-s − 1/3·36-s + 2.43·43-s − 3.50·47-s + 1/7·49-s + 1.41·50-s − 1.10·52-s − 0.133·56-s + 2.04·61-s + 1.01·62-s − 0.251·63-s + 1/8·64-s − 0.977·67-s + 0.235·72-s + ⋯

Functional equation

Λ(s)=(43904s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(43904s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4390443904    =    27⋅732^{7} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 2.799352.79935
Root analytic conductor: 1.293491.29349
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 43904, ( :1/2,1/2), −1)(4,\ 43904,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
7C1C_1 1−T 1 - T
good3C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + 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+4T+pT2)2 ( 1 + 4 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 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47C2C_2 (1+12T+pT2)2 ( 1 + 12 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 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{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

−9.765547119459919407856461234632, −9.649484617797021810103268663298, −8.934540413043951629895618965043, −8.408228795468761224431793416064, −7.68957389720506083723150978195, −7.57571100088867902110310233811, −6.96175265373852036821224883683, −6.17368924384109906924363362759, −5.57928681742950427486583645839, −5.06701824847582369879735343431, −4.31547623044891473234039816601, −3.42443184461883656887541100921, −2.49739457868022289115900838847, −1.90996831082337002056034743405, 0, 1.90996831082337002056034743405, 2.49739457868022289115900838847, 3.42443184461883656887541100921, 4.31547623044891473234039816601, 5.06701824847582369879735343431, 5.57928681742950427486583645839, 6.17368924384109906924363362759, 6.96175265373852036821224883683, 7.57571100088867902110310233811, 7.68957389720506083723150978195, 8.408228795468761224431793416064, 8.934540413043951629895618965043, 9.649484617797021810103268663298, 9.765547119459919407856461234632

Graph of the ZZ-function along the critical line