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Properties

Label 4-350e2-1.1-c3e2-0-11
Degree 44
Conductor 122500122500
Sign 11
Analytic cond. 426.450426.450
Root an. cond. 4.544304.54430
Motivic weight 33
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  − 4·4-s + 5·9-s − 66·11-s + 16·16-s + 140·19-s + 450·29-s − 176·31-s − 20·36-s + 864·41-s + 264·44-s − 49·49-s − 960·59-s + 1.62e3·61-s − 64·64-s + 864·71-s − 560·76-s − 850·79-s − 704·81-s − 1.92e3·89-s − 330·99-s − 876·101-s + 3.83e3·109-s − 1.80e3·116-s + 605·121-s + 704·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/27·9-s − 1.80·11-s + 1/4·16-s + 1.69·19-s + 2.88·29-s − 1.01·31-s − 0.0925·36-s + 3.29·41-s + 0.904·44-s − 1/7·49-s − 2.11·59-s + 3.40·61-s − 1/8·64-s + 1.44·71-s − 0.845·76-s − 1.21·79-s − 0.965·81-s − 2.28·89-s − 0.335·99-s − 0.863·101-s + 3.36·109-s − 1.44·116-s + 5/11·121-s + 0.509·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=(122500s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(122500s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{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: 426.450426.450
Root analytic conductor: 4.544304.54430
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 122500, ( :3/2,3/2), 1)(4,\ 122500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.9425145451.942514545
L(12)L(\frac12) ≈\approx 1.9425145451.942514545
L(52)L(\frac{5}{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+p2T2 1 + p^{2} T^{2}
5 1 1
7C2C_2 1+p2T2 1 + p^{2} T^{2}
good3C22C_2^2 1−5T2+p6T4 1 - 5 T^{2} + p^{6} T^{4}
11C2C_2 (1+3pT+p3T2)2 ( 1 + 3 p T + p^{3} T^{2} )^{2}
13C22C_2^2 1−2545T2+p6T4 1 - 2545 T^{2} + p^{6} T^{4}
17C22C_2^2 1+2495T2+p6T4 1 + 2495 T^{2} + p^{6} T^{4}
19C2C_2 (1−70T+p3T2)2 ( 1 - 70 T + p^{3} T^{2} )^{2}
23C22C_2^2 1−22570T2+p6T4 1 - 22570 T^{2} + p^{6} T^{4}
29C2C_2 (1−225T+p3T2)2 ( 1 - 225 T + p^{3} T^{2} )^{2}
31C2C_2 (1+88T+p3T2)2 ( 1 + 88 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−100150T2+p6T4 1 - 100150 T^{2} + p^{6} T^{4}
41C2C_2 (1−432T+p3T2)2 ( 1 - 432 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−127330T2+p6T4 1 - 127330 T^{2} + p^{6} T^{4}
47C22C_2^2 1−38725T2+p6T4 1 - 38725 T^{2} + p^{6} T^{4}
53C22C_2^2 1+203510T2+p6T4 1 + 203510 T^{2} + p^{6} T^{4}
59C2C_2 (1+480T+p3T2)2 ( 1 + 480 T + p^{3} T^{2} )^{2}
61C2C_2 (1−812T+p3T2)2 ( 1 - 812 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−246310T2+p6T4 1 - 246310 T^{2} + p^{6} T^{4}
71C2C_2 (1−432T+p3T2)2 ( 1 - 432 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−649870T2+p6T4 1 - 649870 T^{2} + p^{6} T^{4}
79C2C_2 (1+425T+p3T2)2 ( 1 + 425 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−198790T2+p6T4 1 - 198790 T^{2} + p^{6} T^{4}
89C2C_2 (1+960T+p3T2)2 ( 1 + 960 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−1322665T2+p6T4 1 - 1322665 T^{2} + p^{6} 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

−11.40985961231806188638372247154, −10.76966040685308886012835996517, −10.24476963401864994185354817956, −9.995752339047791671341062570221, −9.513617207004300963547223004915, −9.017475295201064056858708572042, −8.339658679850305248356127624637, −8.107674714999091627912678035565, −7.45313028963705608528013027553, −7.25563616634511887751888497145, −6.44844333451257791583636340482, −5.76265520629746189179065838504, −5.39350972483079320254787702001, −4.86675750534219107335004466489, −4.36993776404227476984938724357, −3.61767049061658995390571163022, −2.73033807496401543444987247950, −2.59403547171467179085322643698, −1.22113281003954753488476805137, −0.55752770705824908531302636196, 0.55752770705824908531302636196, 1.22113281003954753488476805137, 2.59403547171467179085322643698, 2.73033807496401543444987247950, 3.61767049061658995390571163022, 4.36993776404227476984938724357, 4.86675750534219107335004466489, 5.39350972483079320254787702001, 5.76265520629746189179065838504, 6.44844333451257791583636340482, 7.25563616634511887751888497145, 7.45313028963705608528013027553, 8.107674714999091627912678035565, 8.339658679850305248356127624637, 9.017475295201064056858708572042, 9.513617207004300963547223004915, 9.995752339047791671341062570221, 10.24476963401864994185354817956, 10.76966040685308886012835996517, 11.40985961231806188638372247154

Graph of the ZZ-function along the critical line