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Properties

Label 4-950e2-1.1-c1e2-0-1
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 57.544157.5441
Root an. cond. 2.754232.75423
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 − 3-s + 6-s − 4·7-s + 8-s + 3·9-s − 6·11-s + 6·13-s + 4·14-s − 16-s + 2·17-s − 3·18-s + 7·19-s + 4·21-s + 6·22-s − 8·23-s − 24-s − 6·26-s − 8·27-s + 2·29-s − 16·31-s + 6·33-s − 2·34-s − 16·37-s − 7·38-s − 6·39-s − 5·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 9-s − 1.80·11-s + 1.66·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.60·19-s + 0.872·21-s + 1.27·22-s − 1.66·23-s − 0.204·24-s − 1.17·26-s − 1.53·27-s + 0.371·29-s − 2.87·31-s + 1.04·33-s − 0.342·34-s − 2.63·37-s − 1.13·38-s − 0.960·39-s − 0.780·41-s + ⋯

Functional equation

Λ(s)=(902500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(902500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 902500902500    =    22⋅54⋅1922^{2} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 57.544157.5441
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 902500, ( :1/2,1/2), 1)(4,\ 902500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.16086220910.1608622091
L(12)L(\frac12) ≈\approx 0.16086220910.1608622091
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
19C2C_2 1−7T+pT2 1 - 7 T + p T^{2}
good3C22C_2^2 1+T−2T2+pT3+p2T4 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4}
7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
13C22C_2^2 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−2T−13T2−2pT3+p2T4 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+8T+41T2+8pT3+p2T4 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−2T−25T2−2pT3+p2T4 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4}
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C22C_2^2 1+5T−16T2+5pT3+p2T4 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
47C22C_2^2 1+6T−11T2+6pT3+p2T4 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−6T−17T2−6pT3+p2T4 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+5T−34T2+5pT3+p2T4 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4}
61C2C_2 (1+T+pT2)(1+13T+pT2) ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} )
67C2C_2 (1−11T+pT2)(1+16T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} )
71C22C_2^2 1−6T−35T2−6pT3+p2T4 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+9T+8T2+9pT3+p2T4 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+8T−15T2+8pT3+p2T4 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
89C22C_2^2 1+14T+107T2+14pT3+p2T4 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+15T+128T2+15pT3+p2T4 1 + 15 T + 128 T^{2} + 15 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

−10.36859661640223438742265491932, −9.733230303290900359360727722307, −9.723062748053496413028711491741, −8.925555673642421527702407336663, −8.809603457363235577501433377780, −8.016702852100782720064155637798, −7.80166390321345304139152639750, −7.20960670757774901493112066904, −7.12708564082600825366052882373, −6.31965801127772253482119207790, −6.00537553320877504752555515351, −5.39971952835208150949969959237, −5.35379623129920964278335808955, −4.54638968321674808446313788390, −3.68746108087346518524154159102, −3.47455028564427821478408444633, −3.11755097200733550119288067858, −1.86194946388814138926680489730, −1.55806517875272411449924869470, −0.21899355817458887370701028620, 0.21899355817458887370701028620, 1.55806517875272411449924869470, 1.86194946388814138926680489730, 3.11755097200733550119288067858, 3.47455028564427821478408444633, 3.68746108087346518524154159102, 4.54638968321674808446313788390, 5.35379623129920964278335808955, 5.39971952835208150949969959237, 6.00537553320877504752555515351, 6.31965801127772253482119207790, 7.12708564082600825366052882373, 7.20960670757774901493112066904, 7.80166390321345304139152639750, 8.016702852100782720064155637798, 8.809603457363235577501433377780, 8.925555673642421527702407336663, 9.723062748053496413028711491741, 9.733230303290900359360727722307, 10.36859661640223438742265491932

Graph of the ZZ-function along the critical line