Properties

Label 4-780e2-1.1-c1e2-0-13
Degree $4$
Conductor $608400$
Sign $1$
Analytic cond. $38.7921$
Root an. cond. $2.49566$
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 + 2·4-s + 2·5-s + 9-s + 4·10-s − 2·13-s − 4·16-s − 2·17-s + 2·18-s + 4·20-s + 3·25-s − 4·26-s − 4·29-s − 8·32-s − 4·34-s + 2·36-s + 22·37-s − 10·41-s + 2·45-s − 5·49-s + 6·50-s − 4·52-s + 22·53-s − 8·58-s + 26·61-s − 8·64-s − 4·65-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.894·5-s + 1/3·9-s + 1.26·10-s − 0.554·13-s − 16-s − 0.485·17-s + 0.471·18-s + 0.894·20-s + 3/5·25-s − 0.784·26-s − 0.742·29-s − 1.41·32-s − 0.685·34-s + 1/3·36-s + 3.61·37-s − 1.56·41-s + 0.298·45-s − 5/7·49-s + 0.848·50-s − 0.554·52-s + 3.02·53-s − 1.05·58-s + 3.32·61-s − 64-s − 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(608400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(38.7921\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 608400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.762512499\)
\(L(\frac12)\) \(\approx\) \(4.762512499\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−8.335023576107244243788704360049, −7.972551023048584427904151580263, −7.11315490900767649931627865439, −6.97057067834936316995403518099, −6.52098216309501079985355368795, −5.79375120182502147608870430489, −5.76684274070239466860561733520, −5.05428953135960136559927317103, −4.77834348392706889939521142886, −4.05163184530249924926116818344, −3.83066202020499030341074962519, −3.00135498973141972833429935360, −2.26942722047331000485028011206, −2.21108906834861592943657431933, −0.888327107629161525168347183801, 0.888327107629161525168347183801, 2.21108906834861592943657431933, 2.26942722047331000485028011206, 3.00135498973141972833429935360, 3.83066202020499030341074962519, 4.05163184530249924926116818344, 4.77834348392706889939521142886, 5.05428953135960136559927317103, 5.76684274070239466860561733520, 5.79375120182502147608870430489, 6.52098216309501079985355368795, 6.97057067834936316995403518099, 7.11315490900767649931627865439, 7.972551023048584427904151580263, 8.335023576107244243788704360049

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