Properties

Label 4-4851e2-1.1-c1e2-0-5
Degree $4$
Conductor $23532201$
Sign $1$
Analytic cond. $1500.43$
Root an. cond. $6.22377$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s + 4·5-s + 8·10-s − 2·11-s + 4·13-s + 16-s + 6·17-s + 6·19-s + 4·20-s − 4·22-s + 14·23-s + 2·25-s + 8·26-s + 2·29-s − 2·32-s + 12·34-s − 2·37-s + 12·38-s + 8·41-s − 14·43-s − 2·44-s + 28·46-s − 14·47-s + 4·50-s + 4·52-s + 20·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 1.78·5-s + 2.52·10-s − 0.603·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 1.37·19-s + 0.894·20-s − 0.852·22-s + 2.91·23-s + 2/5·25-s + 1.56·26-s + 0.371·29-s − 0.353·32-s + 2.05·34-s − 0.328·37-s + 1.94·38-s + 1.24·41-s − 2.13·43-s − 0.301·44-s + 4.12·46-s − 2.04·47-s + 0.565·50-s + 0.554·52-s + 2.74·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23532201\)    =    \(3^{4} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1500.43\)
Root analytic conductor: \(6.22377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23532201,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(12.31196884\)
\(L(\frac12)\) \(\approx\) \(12.31196884\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 2 T + 41 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 14 T + 135 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 14 T + 183 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−8.449250601034422348439657264717, −7.987364910507144321020014733148, −7.72858996474091910668025923090, −7.22062686295560240153687977722, −6.79521538585397719645147811432, −6.63016547074526999917319745606, −5.97255803813924492620417901424, −5.76757748163586321867570624091, −5.41406540298674561710547744502, −5.23561173656975590429736390450, −4.89790082122465375778514851087, −4.63414250906547442586579249520, −3.76845253310784830154078976926, −3.61599654611592607405100288231, −3.14110669469976902497524644594, −2.90029162951859411235141727478, −2.27893739810726249470998704493, −1.67635631206165915606167263552, −1.24970897769674826198021931009, −0.820114828312688323800743213161, 0.820114828312688323800743213161, 1.24970897769674826198021931009, 1.67635631206165915606167263552, 2.27893739810726249470998704493, 2.90029162951859411235141727478, 3.14110669469976902497524644594, 3.61599654611592607405100288231, 3.76845253310784830154078976926, 4.63414250906547442586579249520, 4.89790082122465375778514851087, 5.23561173656975590429736390450, 5.41406540298674561710547744502, 5.76757748163586321867570624091, 5.97255803813924492620417901424, 6.63016547074526999917319745606, 6.79521538585397719645147811432, 7.22062686295560240153687977722, 7.72858996474091910668025923090, 7.987364910507144321020014733148, 8.449250601034422348439657264717

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