↑

Properties

Label 4-950e2-1.1-c1e2-0-13
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 6-s + 4·7-s − 8-s + 3·9-s + 6·13-s + 4·14-s − 16-s + 7·17-s + 3·18-s + 7·19-s + 4·21-s + 2·23-s − 24-s + 6·26-s + 8·27-s − 10·29-s − 4·31-s + 7·34-s − 8·37-s + 7·38-s + 6·39-s − 2·41-s + 4·42-s − 12·43-s + 2·46-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 + 1.66·13-s + 1.06·14-s − 1/4·16-s + 1.69·17-s + 0.707·18-s + 1.60·19-s + 0.872·21-s + 0.417·23-s − 0.204·24-s + 1.17·26-s + 1.53·27-s − 1.85·29-s − 0.718·31-s + 1.20·34-s − 1.31·37-s + 1.13·38-s + 0.960·39-s − 0.312·41-s + 0.617·42-s − 1.82·43-s + 0.294·46-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 5.7231717395.723171739
L(12)L(\frac12) ≈\approx 5.7231717395.723171739
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+pT2)2 ( 1 + 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 1−7T+32T2−7pT3+p2T4 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−2T−19T2−2pT3+p2T4 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+10T+71T2+10pT3+p2T4 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4}
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C22C_2^2 1+2T−37T2+2pT3+p2T4 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+12T+101T2+12pT3+p2T4 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
59C22C_2^2 1−T−58T2−pT3+p2T4 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4}
61C22C_2^2 1+8T+3T2+8pT3+p2T4 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−8T−3T2−8pT3+p2T4 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−12T+73T2−12pT3+p2T4 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+3T−64T2+3pT3+p2T4 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4}
79C2C_2 (1−17T+pT2)(1+13T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} )
83C2C_2 (1−13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}
89C22C_2^2 1−13T+80T2−13pT3+p2T4 1 - 13 T + 80 T^{2} - 13 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}
show more
show less
   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

−10.34599039577363938155291269900, −9.753896958038120181784811519343, −9.257148210482455452088647790531, −9.180718358012264093291878714398, −8.395764872650266971956110983382, −8.119250635477307103428660859525, −7.82417661959673157211801893168, −7.43396729172931931625275618412, −6.79156902407378647287619746137, −6.50209472958931857661344709605, −5.68040130349886059775860832322, −5.23211496517706792481320258497, −5.15097657793165521358766752593, −4.58659887787054676319870682825, −3.73999173967527726462976642243, −3.40874993968444127245205612980, −3.39247267870296325394236208298, −2.17986438884057206823037595684, −1.39439241757736951094852567552, −1.27015546854278142233877311079, 1.27015546854278142233877311079, 1.39439241757736951094852567552, 2.17986438884057206823037595684, 3.39247267870296325394236208298, 3.40874993968444127245205612980, 3.73999173967527726462976642243, 4.58659887787054676319870682825, 5.15097657793165521358766752593, 5.23211496517706792481320258497, 5.68040130349886059775860832322, 6.50209472958931857661344709605, 6.79156902407378647287619746137, 7.43396729172931931625275618412, 7.82417661959673157211801893168, 8.119250635477307103428660859525, 8.395764872650266971956110983382, 9.180718358012264093291878714398, 9.257148210482455452088647790531, 9.753896958038120181784811519343, 10.34599039577363938155291269900

Graph of the ZZ-function along the critical line