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Properties

Label 4-1323e2-1.1-c1e2-0-10
Degree 44
Conductor 17503291750329
Sign 11
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 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4-s − 8·8-s + 4·11-s − 7·16-s + 8·22-s + 12·23-s + 6·25-s + 4·29-s + 14·32-s + 6·37-s − 2·43-s − 4·44-s + 24·46-s + 12·50-s − 12·53-s + 8·58-s + 35·64-s + 14·67-s + 12·74-s + 22·79-s − 4·86-s − 32·88-s − 12·92-s − 6·100-s − 24·106-s + 4·107-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s + 1.20·11-s − 7/4·16-s + 1.70·22-s + 2.50·23-s + 6/5·25-s + 0.742·29-s + 2.47·32-s + 0.986·37-s − 0.304·43-s − 0.603·44-s + 3.53·46-s + 1.69·50-s − 1.64·53-s + 1.05·58-s + 35/8·64-s + 1.71·67-s + 1.39·74-s + 2.47·79-s − 0.431·86-s − 3.41·88-s − 1.25·92-s − 3/5·100-s − 2.33·106-s + 0.386·107-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: 11
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: 00
Selberg data: (4, 1750329, ( :1/2,1/2), 1)(4,\ 1750329,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.4535105553.453510555
L(12)L(\frac12) ≈\approx 3.4535105553.453510555
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 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
13C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 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 (1−6T+pT2)2 ( 1 - 6 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 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1+T+pT2)2 ( 1 + 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+6T+pT2)2 ( 1 + 6 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 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
67C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + 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 (1−11T+pT2)2 ( 1 - 11 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 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
97C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 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

−8.067909489204395286784670333572, −7.11526166228041896275075317849, −6.78261162248354124290134429986, −6.37671245558863001404255472344, −6.12925157788915884361808029641, −5.38139354530853751865276991486, −5.05817222407614461034531924213, −4.76511557511489897608661805160, −4.43082498076330884740103768112, −3.72686733644268071328326458986, −3.52186169060206671850993551259, −2.92845750765415073814212919525, −2.54451141824192601627591727152, −1.24852865204000807532187370775, −0.72695453395181431252401129496, 0.72695453395181431252401129496, 1.24852865204000807532187370775, 2.54451141824192601627591727152, 2.92845750765415073814212919525, 3.52186169060206671850993551259, 3.72686733644268071328326458986, 4.43082498076330884740103768112, 4.76511557511489897608661805160, 5.05817222407614461034531924213, 5.38139354530853751865276991486, 6.12925157788915884361808029641, 6.37671245558863001404255472344, 6.78261162248354124290134429986, 7.11526166228041896275075317849, 8.067909489204395286784670333572

Graph of the ZZ-function along the critical line