Properties

Label 4-8820e2-1.1-c1e2-0-5
Degree $4$
Conductor $77792400$
Sign $1$
Analytic cond. $4960.11$
Root an. cond. $8.39214$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·11-s − 10·13-s + 10·17-s + 4·19-s − 4·23-s + 3·25-s − 2·29-s − 12·31-s − 4·37-s + 4·41-s + 12·43-s + 2·47-s + 16·53-s − 4·55-s + 4·59-s − 16·61-s + 20·65-s − 8·67-s + 4·71-s − 16·73-s − 2·79-s − 16·83-s − 20·85-s − 8·95-s − 6·97-s − 4·101-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.603·11-s − 2.77·13-s + 2.42·17-s + 0.917·19-s − 0.834·23-s + 3/5·25-s − 0.371·29-s − 2.15·31-s − 0.657·37-s + 0.624·41-s + 1.82·43-s + 0.291·47-s + 2.19·53-s − 0.539·55-s + 0.520·59-s − 2.04·61-s + 2.48·65-s − 0.977·67-s + 0.474·71-s − 1.87·73-s − 0.225·79-s − 1.75·83-s − 2.16·85-s − 0.820·95-s − 0.609·97-s − 0.398·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(77792400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4960.11\)
Root analytic conductor: \(8.39214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8820} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 77792400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good11$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 10 T + 49 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 16 T + 152 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 100 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T - 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.51506839083669035790704428209, −7.39147256907009360363995972721, −7.10379202550673042638583436948, −6.82565001819282357447601997375, −6.00786672581556770267742002191, −5.79063171936550874462078254278, −5.39548232140894837777117422221, −5.38931518456093638143642162686, −4.67304589246096366563352339139, −4.51371035017607411203142572783, −3.85243241081626781515390275987, −3.84517069573274239588195828617, −3.21147413039260216725818467844, −2.98000595675622198442627654925, −2.38412671210003606905580618141, −2.14755042855335536465212613578, −1.24424481041646592185485850622, −1.13345424901774665842978120655, 0, 0, 1.13345424901774665842978120655, 1.24424481041646592185485850622, 2.14755042855335536465212613578, 2.38412671210003606905580618141, 2.98000595675622198442627654925, 3.21147413039260216725818467844, 3.84517069573274239588195828617, 3.85243241081626781515390275987, 4.51371035017607411203142572783, 4.67304589246096366563352339139, 5.38931518456093638143642162686, 5.39548232140894837777117422221, 5.79063171936550874462078254278, 6.00786672581556770267742002191, 6.82565001819282357447601997375, 7.10379202550673042638583436948, 7.39147256907009360363995972721, 7.51506839083669035790704428209

Graph of the $Z$-function along the critical line