Properties

Label 4-7448e2-1.1-c1e2-0-7
Degree $4$
Conductor $55472704$
Sign $1$
Analytic cond. $3536.98$
Root an. cond. $7.71184$
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  + 4·3-s − 2·5-s + 8·9-s + 2·11-s − 4·13-s − 8·15-s − 8·17-s − 2·19-s + 6·23-s − 7·25-s + 12·27-s − 16·29-s + 12·31-s + 8·33-s − 16·37-s − 16·39-s + 2·43-s − 16·45-s + 6·47-s − 32·51-s − 8·53-s − 4·55-s − 8·57-s + 4·59-s − 10·61-s + 8·65-s + 12·67-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 8/3·9-s + 0.603·11-s − 1.10·13-s − 2.06·15-s − 1.94·17-s − 0.458·19-s + 1.25·23-s − 7/5·25-s + 2.30·27-s − 2.97·29-s + 2.15·31-s + 1.39·33-s − 2.63·37-s − 2.56·39-s + 0.304·43-s − 2.38·45-s + 0.875·47-s − 4.48·51-s − 1.09·53-s − 0.539·55-s − 1.05·57-s + 0.520·59-s − 1.28·61-s + 0.992·65-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55472704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55472704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(55472704\)    =    \(2^{6} \cdot 7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3536.98\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7448} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 55472704,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 16 T + 120 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 16 T + 136 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 120 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 12 T + 162 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 24 T + 268 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 93 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T - 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 162 T^{2} + p^{2} T^{4} \)
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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

−7.68671780457334911703013126390, −7.48273720693119507701024902879, −7.17083149056855568931119015950, −6.98722693210065874210310222721, −6.50174383654703528385856391609, −6.08180599030772492098494538270, −5.62447836255410165664281991454, −5.07988392016100512362776925541, −4.56785945014152545214228065817, −4.45929701660673440662425158889, −3.95647001211056156835654488173, −3.75173963786080742346754762413, −3.19619883423902127383591066751, −3.07867313043376728731086171348, −2.32071188980507502358767702133, −2.31329792265479680080536183013, −1.77882389418539504067161352131, −1.30501510385302863544748041795, 0, 0, 1.30501510385302863544748041795, 1.77882389418539504067161352131, 2.31329792265479680080536183013, 2.32071188980507502358767702133, 3.07867313043376728731086171348, 3.19619883423902127383591066751, 3.75173963786080742346754762413, 3.95647001211056156835654488173, 4.45929701660673440662425158889, 4.56785945014152545214228065817, 5.07988392016100512362776925541, 5.62447836255410165664281991454, 6.08180599030772492098494538270, 6.50174383654703528385856391609, 6.98722693210065874210310222721, 7.17083149056855568931119015950, 7.48273720693119507701024902879, 7.68671780457334911703013126390

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