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Properties

Label 4-1134e2-1.1-c1e2-0-44
Degree 44
Conductor 12859561285956
Sign −1-1
Analytic cond. 81.993681.9936
Root an. cond. 3.009153.00915
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  − 2·2-s + 3·4-s − 4·7-s − 4·8-s + 8·14-s + 5·16-s − 25-s − 12·28-s + 18·29-s − 6·32-s − 2·37-s + 16·43-s + 9·49-s + 2·50-s − 12·53-s + 16·56-s − 36·58-s + 7·64-s − 8·67-s − 24·71-s + 4·74-s − 32·79-s − 32·86-s − 18·98-s − 3·100-s + 24·106-s + 24·107-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.51·7-s − 1.41·8-s + 2.13·14-s + 5/4·16-s − 1/5·25-s − 2.26·28-s + 3.34·29-s − 1.06·32-s − 0.328·37-s + 2.43·43-s + 9/7·49-s + 0.282·50-s − 1.64·53-s + 2.13·56-s − 4.72·58-s + 7/8·64-s − 0.977·67-s − 2.84·71-s + 0.464·74-s − 3.60·79-s − 3.45·86-s − 1.81·98-s − 0.299·100-s + 2.33·106-s + 2.32·107-s + ⋯

Functional equation

Λ(s)=(1285956s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1285956s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12859561285956    =    22⋅38⋅722^{2} \cdot 3^{8} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 81.993681.9936
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1285956, ( :1/2,1/2), −1)(4,\ 1285956,\ (\ :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)2 ( 1 + T )^{2}
3 1 1
7C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good5C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
17C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{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)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
79C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{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

−7.83804568267963926704994118011, −7.23037692302101024142168718583, −7.16277862972928710155421440227, −6.39717818377493347523565938673, −6.29261335708839240986736404399, −5.90045021115349717827393199661, −5.27862826325532129316709233362, −4.37202457039453886575368172384, −4.25638665319375785757127060430, −3.16355858187330818291475189494, −2.94651413139897301734292877776, −2.59312842582062058372859748021, −1.58788487890578855994666762814, −0.901297860986010122087650764417, 0, 0.901297860986010122087650764417, 1.58788487890578855994666762814, 2.59312842582062058372859748021, 2.94651413139897301734292877776, 3.16355858187330818291475189494, 4.25638665319375785757127060430, 4.37202457039453886575368172384, 5.27862826325532129316709233362, 5.90045021115349717827393199661, 6.29261335708839240986736404399, 6.39717818377493347523565938673, 7.16277862972928710155421440227, 7.23037692302101024142168718583, 7.83804568267963926704994118011

Graph of the ZZ-function along the critical line