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Properties

Label 4-350e2-1.1-c1e2-0-8
Degree 44
Conductor 122500122500
Sign 11
Analytic cond. 7.810707.81070
Root an. cond. 1.671751.67175
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 − 7-s − 8-s + 3·9-s + 2·11-s − 14-s − 16-s − 7·17-s + 3·18-s + 2·22-s + 3·23-s + 12·29-s + 7·31-s − 7·34-s − 4·37-s − 14·41-s + 16·43-s + 3·46-s − 7·47-s − 6·49-s + 4·53-s + 56-s + 12·58-s + 14·59-s + 14·61-s + 7·62-s − 3·63-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.377·7-s − 0.353·8-s + 9-s + 0.603·11-s − 0.267·14-s − 1/4·16-s − 1.69·17-s + 0.707·18-s + 0.426·22-s + 0.625·23-s + 2.22·29-s + 1.25·31-s − 1.20·34-s − 0.657·37-s − 2.18·41-s + 2.43·43-s + 0.442·46-s − 1.02·47-s − 6/7·49-s + 0.549·53-s + 0.133·56-s + 1.57·58-s + 1.82·59-s + 1.79·61-s + 0.889·62-s − 0.377·63-s + ⋯

Functional equation

Λ(s)=(122500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(122500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 122500122500    =    22⋅54⋅722^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 7.810707.81070
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 122500, ( :1/2,1/2), 1)(4,\ 122500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2170632192.217063219
L(12)L(\frac12) ≈\approx 2.2170632192.217063219
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}
5 1 1
7C2C_2 1+T+pT2 1 + T + p T^{2}
good3C2C_2 (1−pT+pT2)(1+pT+pT2) ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )
11C22C_2^2 1−2T−7T2−2pT3+p2T4 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C22C_2^2 1+7T+32T2+7pT3+p2T4 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
23C22C_2^2 1−3T−14T2−3pT3+p2T4 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1−11T+pT2)(1+4T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C22C_2^2 1+4T−21T2+4pT3+p2T4 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4}
41C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47C22C_2^2 1+7T+2T2+7pT3+p2T4 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−4T−37T2−4pT3+p2T4 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−14T+137T2−14pT3+p2T4 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4}
61C2C_2 (1−13T+pT2)(1−T+pT2) ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} )
67C22C_2^2 1−12T+77T2−12pT3+p2T4 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4}
71C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
73C22C_2^2 1−14T+123T2−14pT3+p2T4 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−11T+42T2−11pT3+p2T4 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4}
83C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
89C22C_2^2 1+7T−40T2+7pT3+p2T4 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4}
97C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{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

−11.86829101099404790262712064987, −11.30186564908633296258363199411, −10.91873840622636396596505910858, −10.26103845395232419857560481787, −9.828613902065009203468454290904, −9.656785970563092861908815034313, −8.675316991609847082404949933469, −8.646716636315804039807457408765, −8.116192448671307693468838603486, −7.07327476270978309141219495726, −6.80991542849221230349033387897, −6.60121912851246192474476526756, −5.96132472153781885865701835896, −4.96516076979752092931154537741, −4.88489845734171203037522675899, −4.10702267231352623105337212255, −3.76230548808931036220762617250, −2.85248647170399003682824530588, −2.20496407227810848497295250088, −1.00315219492347088514687749877, 1.00315219492347088514687749877, 2.20496407227810848497295250088, 2.85248647170399003682824530588, 3.76230548808931036220762617250, 4.10702267231352623105337212255, 4.88489845734171203037522675899, 4.96516076979752092931154537741, 5.96132472153781885865701835896, 6.60121912851246192474476526756, 6.80991542849221230349033387897, 7.07327476270978309141219495726, 8.116192448671307693468838603486, 8.646716636315804039807457408765, 8.675316991609847082404949933469, 9.656785970563092861908815034313, 9.828613902065009203468454290904, 10.26103845395232419857560481787, 10.91873840622636396596505910858, 11.30186564908633296258363199411, 11.86829101099404790262712064987

Graph of the ZZ-function along the critical line