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Properties

Label 4-1134e2-1.1-c1e2-0-30
Degree 44
Conductor 12859561285956
Sign 11
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 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 6·5-s − 4·7-s − 8-s − 6·10-s + 6·11-s − 2·13-s − 4·14-s − 16-s + 6·17-s − 2·19-s + 6·22-s + 12·23-s + 17·25-s − 2·26-s + 9·29-s + 7·31-s + 6·34-s + 24·35-s + 10·37-s − 2·38-s + 6·40-s + 4·43-s + 12·46-s + 12·47-s + 9·49-s + 17·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.68·5-s − 1.51·7-s − 0.353·8-s − 1.89·10-s + 1.80·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s + 1.27·22-s + 2.50·23-s + 17/5·25-s − 0.392·26-s + 1.67·29-s + 1.25·31-s + 1.02·34-s + 4.05·35-s + 1.64·37-s − 0.324·38-s + 0.948·40-s + 0.609·43-s + 1.76·46-s + 1.75·47-s + 9/7·49-s + 2.40·50-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: 11
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: 00
Selberg data: (4, 1285956, ( :1/2,1/2), 1)(4,\ 1285956,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6906621461.690662146
L(12)L(\frac12) ≈\approx 1.6906621461.690662146
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)
bad2C2C_2 1−T+T2 1 - T + T^{2}
3 1 1
7C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C2C_2 (1−5T+pT2)(1+7T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C22C_2^2 1−6T+19T2−6pT3+p2T4 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29C22C_2^2 1−9T+52T2−9pT3+p2T4 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4}
31C2C_2 (1−11T+pT2)(1+4T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−11T+pT2)(1+T+pT2) ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} )
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C22C_2^2 1−4T−27T2−4pT3+p2T4 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−12T+97T2−12pT3+p2T4 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+3T−44T2+3pT3+p2T4 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+3T−50T2+3pT3+p2T4 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−4T−45T2−4pT3+p2T4 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+2T−63T2+2pT3+p2T4 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1+2T−69T2+2pT3+p2T4 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+5T−54T2+5pT3+p2T4 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−9T−2T2−9pT3+p2T4 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+6T−53T2+6pT3+p2T4 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−13T+72T2−13pT3+p2T4 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4}
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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

−9.787139695390997440403493160242, −9.726054653558983734764329727628, −9.070680312802161235305342108040, −8.851647777639541928662659320889, −8.353231050658178692623954606532, −7.85397161044719272567085924444, −7.52661575484205842675603170925, −7.06952485579367031972517453728, −6.63261219628726278398161055772, −6.48825635455924513510815005127, −5.81213056561974991646725969432, −5.21324057022628905359156653046, −4.49991948914571717453111671184, −4.24487051295829938467574555157, −4.02488429265996079437928500239, −3.38669133562200595296484428438, −2.96768067327223288966911017286, −2.82458963834384751437891636840, −0.975484638579017762636638776897, −0.70598724177145393018985165130, 0.70598724177145393018985165130, 0.975484638579017762636638776897, 2.82458963834384751437891636840, 2.96768067327223288966911017286, 3.38669133562200595296484428438, 4.02488429265996079437928500239, 4.24487051295829938467574555157, 4.49991948914571717453111671184, 5.21324057022628905359156653046, 5.81213056561974991646725969432, 6.48825635455924513510815005127, 6.63261219628726278398161055772, 7.06952485579367031972517453728, 7.52661575484205842675603170925, 7.85397161044719272567085924444, 8.353231050658178692623954606532, 8.851647777639541928662659320889, 9.070680312802161235305342108040, 9.726054653558983734764329727628, 9.787139695390997440403493160242

Graph of the ZZ-function along the critical line