Properties

Label 4-1064e2-1.1-c1e2-0-25
Degree $4$
Conductor $1132096$
Sign $1$
Analytic cond. $72.1834$
Root an. cond. $2.91480$
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  − 3-s − 2·5-s − 2·7-s − 2·9-s − 11-s + 2·15-s + 7·17-s + 2·19-s + 2·21-s − 8·23-s − 7·25-s + 2·27-s − 5·29-s − 13·31-s + 33-s + 4·35-s − 3·41-s + 4·45-s − 8·47-s + 3·49-s − 7·51-s + 3·53-s + 2·55-s − 2·57-s + 4·59-s + 2·61-s + 4·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s + 0.516·15-s + 1.69·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s − 7/5·25-s + 0.384·27-s − 0.928·29-s − 2.33·31-s + 0.174·33-s + 0.676·35-s − 0.468·41-s + 0.596·45-s − 1.16·47-s + 3/7·49-s − 0.980·51-s + 0.412·53-s + 0.269·55-s − 0.264·57-s + 0.520·59-s + 0.256·61-s + 0.503·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1132096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1132096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1132096\)    =    \(2^{6} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(72.1834\)
Root analytic conductor: \(2.91480\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1132096,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 61 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 13 T + 101 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 81 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 79 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 109 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 71 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 11 T + 135 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 29 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T + 117 T^{2} + p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 3 T + 87 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 113 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−9.745117359565659357242330985903, −9.463460236620376338393224653993, −8.663661194119957934342112649897, −8.603365665371041337773328325363, −7.73644267021588761653508413257, −7.67202475509365774400635116925, −7.41163122871988559973614870312, −6.77689653930607795693004811791, −5.98612541146469300117065519906, −5.93038822236590542551801023854, −5.45696108401680629619920110978, −5.18202102813611815477541820396, −4.16623060970997008570478854673, −3.93945824571502802441167542365, −3.38522443720405232099882162259, −3.08023459896184746060602119412, −2.13950487143381518025975937095, −1.47341862756899216645521327926, 0, 0, 1.47341862756899216645521327926, 2.13950487143381518025975937095, 3.08023459896184746060602119412, 3.38522443720405232099882162259, 3.93945824571502802441167542365, 4.16623060970997008570478854673, 5.18202102813611815477541820396, 5.45696108401680629619920110978, 5.93038822236590542551801023854, 5.98612541146469300117065519906, 6.77689653930607795693004811791, 7.41163122871988559973614870312, 7.67202475509365774400635116925, 7.73644267021588761653508413257, 8.603365665371041337773328325363, 8.663661194119957934342112649897, 9.463460236620376338393224653993, 9.745117359565659357242330985903

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