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Properties

Label 4-1323e2-1.1-c1e2-0-17
Degree 44
Conductor 17503291750329
Sign −1-1
Analytic cond. 111.602111.602
Root an. cond. 3.250263.25026
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 − 4-s + 8·8-s + 10·11-s − 7·16-s − 20·22-s + 2·23-s − 25-s − 4·29-s − 14·32-s + 6·37-s + 12·43-s − 10·44-s − 4·46-s + 2·50-s − 16·53-s + 8·58-s + 35·64-s − 28·67-s − 14·71-s − 12·74-s − 20·79-s − 24·86-s + 80·88-s − 2·92-s + 100-s + 32·106-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 2.82·8-s + 3.01·11-s − 7/4·16-s − 4.26·22-s + 0.417·23-s − 1/5·25-s − 0.742·29-s − 2.47·32-s + 0.986·37-s + 1.82·43-s − 1.50·44-s − 0.589·46-s + 0.282·50-s − 2.19·53-s + 1.05·58-s + 35/8·64-s − 3.42·67-s − 1.66·71-s − 1.39·74-s − 2.25·79-s − 2.58·86-s + 8.52·88-s − 0.208·92-s + 1/10·100-s + 3.10·106-s + ⋯

Functional equation

Λ(s)=(1750329s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1750329s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 17503291750329    =    36⋅743^{6} \cdot 7^{4}
Sign: −1-1
Analytic conductor: 111.602111.602
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1750329, ( :1/2,1/2), −1)(4,\ 1750329,\ (\ :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)
bad3 1 1
7 1 1
good2C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
5C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
23C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
37C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
41C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
43C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
47C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
71C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{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 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
97C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 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

−7.59315018372019842769294095239, −7.28342071225309798898505652310, −7.13167564560012152294416600100, −6.34215171511087571525667957813, −5.96735414787932054483964871891, −5.63752173484997695290532171122, −4.60087511347345218102319658198, −4.47191624657260288271626711721, −4.22351345808096997888083312699, −3.59632548449757607506915649618, −3.08850093557224289960924465680, −1.92610841189085090066739675799, −1.32164798123200269811823069632, −1.12292208190814267208777122415, 0, 1.12292208190814267208777122415, 1.32164798123200269811823069632, 1.92610841189085090066739675799, 3.08850093557224289960924465680, 3.59632548449757607506915649618, 4.22351345808096997888083312699, 4.47191624657260288271626711721, 4.60087511347345218102319658198, 5.63752173484997695290532171122, 5.96735414787932054483964871891, 6.34215171511087571525667957813, 7.13167564560012152294416600100, 7.28342071225309798898505652310, 7.59315018372019842769294095239

Graph of the ZZ-function along the critical line