Properties

Label 4-4840e2-1.1-c1e2-0-6
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s − 2·7-s − 9-s − 4·13-s + 4·15-s − 4·19-s − 4·21-s − 4·23-s + 3·25-s − 6·27-s − 4·29-s − 4·35-s − 8·37-s − 8·39-s + 2·41-s − 10·43-s − 2·45-s + 18·47-s − 9·49-s − 4·53-s − 8·57-s − 2·61-s + 2·63-s − 8·65-s − 10·67-s − 8·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 0.755·7-s − 1/3·9-s − 1.10·13-s + 1.03·15-s − 0.917·19-s − 0.872·21-s − 0.834·23-s + 3/5·25-s − 1.15·27-s − 0.742·29-s − 0.676·35-s − 1.31·37-s − 1.28·39-s + 0.312·41-s − 1.52·43-s − 0.298·45-s + 2.62·47-s − 9/7·49-s − 0.549·53-s − 1.05·57-s − 0.256·61-s + 0.251·63-s − 0.992·65-s − 1.22·67-s − 0.963·69-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: \(2\)
Selberg data: \((4,\ 23425600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 18 T + 173 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 10 T + 157 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−8.058279995326508253655814659964, −7.990081923723475962359538034694, −7.23225960187158788236495813497, −7.17014427154616242306000309402, −6.60791223161000068934344203031, −6.37589417543772407538455416765, −5.87177643209506898047569420960, −5.56882464552474528920495380947, −5.27292115474471165422006027930, −4.76928009499290836110007521683, −4.17295683985551882202316470834, −3.99851260173413111561598172215, −3.23708800964699973850055852582, −3.15289610870511843365239976533, −2.48067161897201629187424536431, −2.43266768257858224487519950451, −1.79458197211543779666533084517, −1.38373503648434482203986842830, 0, 0, 1.38373503648434482203986842830, 1.79458197211543779666533084517, 2.43266768257858224487519950451, 2.48067161897201629187424536431, 3.15289610870511843365239976533, 3.23708800964699973850055852582, 3.99851260173413111561598172215, 4.17295683985551882202316470834, 4.76928009499290836110007521683, 5.27292115474471165422006027930, 5.56882464552474528920495380947, 5.87177643209506898047569420960, 6.37589417543772407538455416765, 6.60791223161000068934344203031, 7.17014427154616242306000309402, 7.23225960187158788236495813497, 7.990081923723475962359538034694, 8.058279995326508253655814659964

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