Properties

Label 4-300e2-1.1-c5e2-0-4
Degree 44
Conductor 9000090000
Sign 11
Analytic cond. 2315.062315.06
Root an. cond. 6.936506.93650
Motivic weight 55
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  − 81·9-s + 432·11-s − 3.16e3·19-s + 9.13e3·29-s + 5.48e3·31-s − 2.67e4·41-s + 3.16e4·49-s + 3.11e4·59-s + 7.86e4·61-s + 1.14e5·71-s + 2.11e4·79-s + 6.56e3·81-s + 2.32e5·89-s − 3.49e4·99-s + 5.52e4·101-s + 4.60e5·109-s − 1.82e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.49e5·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.07·11-s − 2.00·19-s + 2.01·29-s + 1.02·31-s − 2.48·41-s + 1.88·49-s + 1.16·59-s + 2.70·61-s + 2.68·71-s + 0.380·79-s + 1/9·81-s + 3.11·89-s − 0.358·99-s + 0.538·101-s + 3.71·109-s − 1.13·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.403·169-s + ⋯

Functional equation

Λ(s)=(90000s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(90000s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9000090000    =    2432542^{4} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 2315.062315.06
Root analytic conductor: 6.936506.93650
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 90000, ( :5/2,5/2), 1)(4,\ 90000,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) \approx 3.1291214563.129121456
L(12)L(\frac12) \approx 3.1291214563.129121456
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C2C_2 1+p4T2 1 + p^{4} T^{2}
5 1 1
good7C22C_2^2 131678T2+p10T4 1 - 31678 T^{2} + p^{10} T^{4}
11C2C_2 (1216T+p5T2)2 ( 1 - 216 T + p^{5} T^{2} )^{2}
13C22C_2^2 1149686T2+p10T4 1 - 149686 T^{2} + p^{10} T^{4}
17C22C_2^2 12554558T2+p10T4 1 - 2554558 T^{2} + p^{10} T^{4}
19C2C_2 (1+1580T+p5T2)2 ( 1 + 1580 T + p^{5} T^{2} )^{2}
23C22C_2^2 14439470T2+p10T4 1 - 4439470 T^{2} + p^{10} T^{4}
29C2C_2 (14566T+p5T2)2 ( 1 - 4566 T + p^{5} T^{2} )^{2}
31C2C_2 (12744T+p5T2)2 ( 1 - 2744 T + p^{5} T^{2} )^{2}
37C22C_2^2 1136608550T2+p10T4 1 - 136608550 T^{2} + p^{10} T^{4}
41C2C_2 (1+13350T+p5T2)2 ( 1 + 13350 T + p^{5} T^{2} )^{2}
43C22C_2^2 1+1960730T2+p10T4 1 + 1960730 T^{2} + p^{10} T^{4}
47C22C_2^2 1341531038T2+p10T4 1 - 341531038 T^{2} + p^{10} T^{4}
53C22C_2^2 1737547622T2+p10T4 1 - 737547622 T^{2} + p^{10} T^{4}
59C2C_2 (1264pT+p5T2)2 ( 1 - 264 p T + p^{5} T^{2} )^{2}
61C2C_2 (139302T+p5T2)2 ( 1 - 39302 T + p^{5} T^{2} )^{2}
67C22C_2^2 1+412943402T2+p10T4 1 + 412943402 T^{2} + p^{10} T^{4}
71C2C_2 (157120T+p5T2)2 ( 1 - 57120 T + p^{5} T^{2} )^{2}
73C22C_2^2 11605781582T2+p10T4 1 - 1605781582 T^{2} + p^{10} T^{4}
79C2C_2 (110552T+p5T2)2 ( 1 - 10552 T + p^{5} T^{2} )^{2}
83C22C_2^2 1+567290p2T2+p10T4 1 + 567290 p^{2} T^{2} + p^{10} T^{4}
89C2C_2 (1116430T+p5T2)2 ( 1 - 116430 T + p^{5} T^{2} )^{2}
97C22C_2^2 117166940990T2+p10T4 1 - 17166940990 T^{2} + p^{10} T^{4}
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   L(s)=p j=14(1αj,pps)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

−11.21115750836384368434329691411, −10.53104170324327817908202198584, −10.17441185710974251919933355535, −9.959430315134498845067613761964, −9.042912236779925476228196944217, −8.768100900965576323745367955383, −8.299413736454956642585167024553, −8.068771507561092723159468306081, −6.93576742330407891727936038341, −6.79751272198149287838526002146, −6.33720193390784163211742803727, −5.80699508266202579623123524944, −4.86707540992403090175228484694, −4.71558616395278519890037109159, −3.73206934808178501428819469534, −3.58330081840341827902651458639, −2.33615532441914227899475679760, −2.21704720789067257294825913843, −1.03548875318616564790633239603, −0.56064957615016467624518410478, 0.56064957615016467624518410478, 1.03548875318616564790633239603, 2.21704720789067257294825913843, 2.33615532441914227899475679760, 3.58330081840341827902651458639, 3.73206934808178501428819469534, 4.71558616395278519890037109159, 4.86707540992403090175228484694, 5.80699508266202579623123524944, 6.33720193390784163211742803727, 6.79751272198149287838526002146, 6.93576742330407891727936038341, 8.068771507561092723159468306081, 8.299413736454956642585167024553, 8.768100900965576323745367955383, 9.042912236779925476228196944217, 9.959430315134498845067613761964, 10.17441185710974251919933355535, 10.53104170324327817908202198584, 11.21115750836384368434329691411

Graph of the ZZ-function along the critical line