Properties

Label 4-4840e2-1.1-c1e2-0-0
Degree $4$
Conductor $23425600$
Sign $1$
Analytic cond. $1493.63$
Root an. cond. $6.21671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 5·7-s − 9-s − 6·13-s − 2·15-s + 3·17-s − 7·19-s − 5·21-s − 6·23-s + 3·25-s − 13·29-s − 7·31-s + 10·35-s + 19·37-s − 6·39-s + 8·41-s − 2·43-s + 2·45-s + 2·47-s + 9·49-s + 3·51-s + 7·53-s − 7·57-s − 6·59-s + 21·61-s + 5·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1.88·7-s − 1/3·9-s − 1.66·13-s − 0.516·15-s + 0.727·17-s − 1.60·19-s − 1.09·21-s − 1.25·23-s + 3/5·25-s − 2.41·29-s − 1.25·31-s + 1.69·35-s + 3.12·37-s − 0.960·39-s + 1.24·41-s − 0.304·43-s + 0.298·45-s + 0.291·47-s + 9/7·49-s + 0.420·51-s + 0.961·53-s − 0.927·57-s − 0.781·59-s + 2.68·61-s + 0.629·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23425600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23425600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23425600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1493.63\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23425600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3044894943\)
\(L(\frac12)\) \(\approx\) \(0.3044894943\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 + 7 T + 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 19 T + 160 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T - 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 7 T + 80 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 21 T + 228 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
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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

−8.192867505434845688401446029270, −8.176882668763962362592242359863, −7.67278783161813013540747539820, −7.48044850485874261910178722568, −7.07351747991955619339429976468, −6.76168284624414686379666423102, −6.16149101367801538889825010866, −6.11020780637477561677481686048, −5.47449664398454380934839962458, −5.36616737059520996968221228290, −4.56487870403462412177180340379, −4.04834404435399700897344942493, −4.03951597220707938161766854999, −3.59446625467074219122793148662, −3.04590249400907629988524953309, −2.71954339328503142037862772597, −2.23907318683614832488887099808, −2.01208041590450070263547837715, −0.804036707494051472523679283186, −0.18442254081183993471636535198, 0.18442254081183993471636535198, 0.804036707494051472523679283186, 2.01208041590450070263547837715, 2.23907318683614832488887099808, 2.71954339328503142037862772597, 3.04590249400907629988524953309, 3.59446625467074219122793148662, 4.03951597220707938161766854999, 4.04834404435399700897344942493, 4.56487870403462412177180340379, 5.36616737059520996968221228290, 5.47449664398454380934839962458, 6.11020780637477561677481686048, 6.16149101367801538889825010866, 6.76168284624414686379666423102, 7.07351747991955619339429976468, 7.48044850485874261910178722568, 7.67278783161813013540747539820, 8.176882668763962362592242359863, 8.192867505434845688401446029270

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