Properties

Label 4-3920e2-1.1-c0e2-0-1
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $3.82724$
Root an. cond. $1.39869$
Motivic weight $0$
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  − 2·9-s − 25-s − 4·29-s + 3·81-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + 227-s + ⋯
L(s)  = 1  − 2·9-s − 25-s − 4·29-s + 3·81-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.82724\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5680167293\)
\(L(\frac12)\) \(\approx\) \(0.5680167293\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$ \( ( 1 + T )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−8.839457742917624730236885057138, −8.368707523433423459779826700280, −8.288877930214538820440036348875, −7.53641197172638003296870852757, −7.47101949045727111959641763989, −7.26430085416523957860440244634, −6.46017603534259207123998851816, −6.08527879990415213721264593576, −5.87953827448520292808163206630, −5.60599793423335018263898031548, −5.13493417296452068192629447385, −4.87260657342359490919098493428, −4.13251387569202791887804554781, −3.72727829830711425869347741413, −3.44228616225946584095229912673, −3.06315498666256162455777069828, −2.24684707486381082344021250918, −2.19577350563495235354461058585, −1.52752880183226603916270052959, −0.40325904393571257818145691232, 0.40325904393571257818145691232, 1.52752880183226603916270052959, 2.19577350563495235354461058585, 2.24684707486381082344021250918, 3.06315498666256162455777069828, 3.44228616225946584095229912673, 3.72727829830711425869347741413, 4.13251387569202791887804554781, 4.87260657342359490919098493428, 5.13493417296452068192629447385, 5.60599793423335018263898031548, 5.87953827448520292808163206630, 6.08527879990415213721264593576, 6.46017603534259207123998851816, 7.26430085416523957860440244634, 7.47101949045727111959641763989, 7.53641197172638003296870852757, 8.288877930214538820440036348875, 8.368707523433423459779826700280, 8.839457742917624730236885057138

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