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Properties

Label 4-3150e2-1.1-c1e2-0-21
Degree 44
Conductor 99225009922500
Sign 11
Analytic cond. 632.667632.667
Root an. cond. 5.015265.01526
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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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)=(9922500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9922500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 99225009922500    =    22⋅34⋅54⋅722^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 632.667632.667
Root analytic conductor: 5.015265.01526
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 9922500, ( :1/2,1/2), 1)(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 8.8180380268.818038026
L(12)L(\frac12) ≈\approx 8.8180380268.818038026
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}
3 1 1
5 1 1
7C1C_1 (1+T)2 ( 1 + T )^{2}
good11C22C_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}
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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

−8.718500198785599827768234947999, −8.671501638574013276719841265501, −7.888616375787963028727798733286, −7.60042568897851394711821255594, −7.30888466539120662197892038024, −6.90059599289980208404241258947, −6.32809171192839689921668176380, −6.30278614976401395409516296359, −5.81661532101620633712824348602, −5.45890174961157953891347456045, −4.97905503640449219249066760321, −4.68742545986080523005139201866, −4.00625611932096696610024124243, −3.88434872757924766026719053158, −3.38127715336835055545745174316, −2.94152371679574146245178227398, −2.49277899131395804471994601550, −2.12373740570617103515999382046, −1.05784179025911638201657044668, −0.886049184299352588669362709967, 0.886049184299352588669362709967, 1.05784179025911638201657044668, 2.12373740570617103515999382046, 2.49277899131395804471994601550, 2.94152371679574146245178227398, 3.38127715336835055545745174316, 3.88434872757924766026719053158, 4.00625611932096696610024124243, 4.68742545986080523005139201866, 4.97905503640449219249066760321, 5.45890174961157953891347456045, 5.81661532101620633712824348602, 6.30278614976401395409516296359, 6.32809171192839689921668176380, 6.90059599289980208404241258947, 7.30888466539120662197892038024, 7.60042568897851394711821255594, 7.888616375787963028727798733286, 8.671501638574013276719841265501, 8.718500198785599827768234947999

Graph of the ZZ-function along the critical line