↑

Properties

Label 4-350e2-1.1-c1e2-0-6
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 2·7-s − 4·8-s + 4·13-s + 4·14-s + 5·16-s + 4·17-s + 8·19-s − 4·23-s − 8·26-s − 6·28-s + 4·29-s + 8·31-s − 6·32-s − 8·34-s + 4·37-s − 16·38-s − 12·41-s + 8·43-s + 8·46-s + 8·47-s + 3·49-s + 12·52-s − 12·53-s + 8·56-s − 8·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s + 1.10·13-s + 1.06·14-s + 5/4·16-s + 0.970·17-s + 1.83·19-s − 0.834·23-s − 1.56·26-s − 1.13·28-s + 0.742·29-s + 1.43·31-s − 1.06·32-s − 1.37·34-s + 0.657·37-s − 2.59·38-s − 1.87·41-s + 1.21·43-s + 1.17·46-s + 1.16·47-s + 3/7·49-s + 1.66·52-s − 1.64·53-s + 1.06·56-s − 1.05·58-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 0.82747903150.8274790315
L(12)L(\frac12) ≈\approx 0.82747903150.8274790315
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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
5 1 1
7C1C_1 (1+T)2 ( 1 + T )^{2}
good3C22C_2^2 1+p2T4 1 + p^{2} T^{4}
11C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13D4D_{4} 1−4T+24T2−4pT3+p2T4 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19D4D_{4} 1−8T+48T2−8pT3+p2T4 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+4T+26T2+4pT3+p2T4 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−4T+38T2−4pT3+p2T4 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−8T+54T2−8pT3+p2T4 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41D4D_{4} 1+12T+94T2+12pT3+p2T4 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−8T+78T2−8pT3+p2T4 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−8T+86T2−8pT3+p2T4 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+12T+118T2+12pT3+p2T4 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+8T+128T2+8pT3+p2T4 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−12T+152T2−12pT3+p2T4 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4}
67C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
71D4D_{4} 1+12T+154T2+12pT3+p2T4 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+4T+126T2+4pT3+p2T4 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−4T+138T2−4pT3+p2T4 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4}
83C22C_2^2 1+160T2+p2T4 1 + 160 T^{2} + p^{2} T^{4}
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97D4D_{4} 1−12T+134T2−12pT3+p2T4 1 - 12 T + 134 T^{2} - 12 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

−11.54672523315593664549491618870, −11.29554923076180718025028882069, −10.55325367837767017951933665498, −10.07728113875637408344071753299, −10.01439033783112463655606599604, −9.485146979364800771182089474166, −8.834744037874522940583574844152, −8.664179296918683319137492382330, −7.86879656006639784858071167283, −7.70971377206101767237854140637, −7.10347793159328129078698097105, −6.53697423851433080849603310003, −5.87882779279032724825489227814, −5.83104606741587686350813742208, −4.80557649010691109432273744906, −3.95932668971741598004652805776, −3.05424496039282091499674828238, −2.97562386654069150960632236205, −1.61740142176884504055417047922, −0.862503666228635144429925468519, 0.862503666228635144429925468519, 1.61740142176884504055417047922, 2.97562386654069150960632236205, 3.05424496039282091499674828238, 3.95932668971741598004652805776, 4.80557649010691109432273744906, 5.83104606741587686350813742208, 5.87882779279032724825489227814, 6.53697423851433080849603310003, 7.10347793159328129078698097105, 7.70971377206101767237854140637, 7.86879656006639784858071167283, 8.664179296918683319137492382330, 8.834744037874522940583574844152, 9.485146979364800771182089474166, 10.01439033783112463655606599604, 10.07728113875637408344071753299, 10.55325367837767017951933665498, 11.29554923076180718025028882069, 11.54672523315593664549491618870

Graph of the ZZ-function along the critical line