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Properties

Label 4-640e2-1.1-c1e2-0-49
Degree 44
Conductor 409600409600
Sign −1-1
Analytic cond. 26.116426.1164
Root an. cond. 2.260622.26062
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 2·5-s + 6·9-s − 4·13-s − 8·15-s − 25-s − 4·27-s − 20·37-s − 16·39-s − 12·41-s + 12·43-s − 12·45-s + 2·49-s + 12·53-s + 8·65-s + 20·67-s − 24·71-s − 4·75-s + 16·79-s − 37·81-s − 12·83-s − 4·89-s − 4·107-s − 80·111-s − 24·117-s − 18·121-s − 48·123-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 2·9-s − 1.10·13-s − 2.06·15-s − 1/5·25-s − 0.769·27-s − 3.28·37-s − 2.56·39-s − 1.87·41-s + 1.82·43-s − 1.78·45-s + 2/7·49-s + 1.64·53-s + 0.992·65-s + 2.44·67-s − 2.84·71-s − 0.461·75-s + 1.80·79-s − 4.11·81-s − 1.31·83-s − 0.423·89-s − 0.386·107-s − 7.59·111-s − 2.21·117-s − 1.63·121-s − 4.32·123-s + ⋯

Functional equation

Λ(s)=(409600s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(409600s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 409600409600    =    214⋅522^{14} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 26.116426.1164
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 409600, ( :1/2,1/2), −1)(4,\ 409600,\ (\ :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)
bad2 1 1
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good3C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{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)2 ( 1 + 2 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{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)2 ( 1 + 10 T + p T^{2} )^{2}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
47C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{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)2 ( 1 - 10 T + p T^{2} )^{2}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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

−8.499715772662989764702573299468, −8.098681382024412731658064042949, −7.48649464154346783477619856196, −7.33738117792255002879146989083, −6.95023759239239106595773853946, −6.15302264435441050088112742528, −5.26498006511685590910060560318, −5.16542238567969836254694300670, −4.04799601522778258065319452547, −3.91058908136761919842529082491, −3.35101374838703231470604183982, −2.81830188953599087445367836313, −2.30676446591740616355718333113, −1.71401499382330216031350685870, 0, 1.71401499382330216031350685870, 2.30676446591740616355718333113, 2.81830188953599087445367836313, 3.35101374838703231470604183982, 3.91058908136761919842529082491, 4.04799601522778258065319452547, 5.16542238567969836254694300670, 5.26498006511685590910060560318, 6.15302264435441050088112742528, 6.95023759239239106595773853946, 7.33738117792255002879146989083, 7.48649464154346783477619856196, 8.098681382024412731658064042949, 8.499715772662989764702573299468

Graph of the ZZ-function along the critical line