Properties

Label 4-4600e2-1.1-c1e2-0-4
Degree $4$
Conductor $21160000$
Sign $1$
Analytic cond. $1349.17$
Root an. cond. $6.06062$
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  + 6·9-s − 12·11-s + 12·19-s − 6·29-s − 6·31-s + 18·41-s + 13·49-s − 2·59-s + 16·61-s − 10·71-s + 27·81-s − 8·89-s − 72·99-s − 18·101-s + 12·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 72·171-s + ⋯
L(s)  = 1  + 2·9-s − 3.61·11-s + 2.75·19-s − 1.11·29-s − 1.07·31-s + 2.81·41-s + 13/7·49-s − 0.260·59-s + 2.04·61-s − 1.18·71-s + 3·81-s − 0.847·89-s − 7.23·99-s − 1.79·101-s + 1.14·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 5.50·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21160000\)    =    \(2^{6} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1349.17\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.667481728\)
\(L(\frac12)\) \(\approx\) \(2.667481728\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−8.332500620763886995163268329317, −7.974684030038371362661499655364, −7.50760568985551690456235254807, −7.46721978945700448306441662723, −7.32455680407539074097898174167, −7.05327383951325524737312396510, −6.23306749822150298562826978274, −5.70561057142759781889338646595, −5.42755694932790245043144064116, −5.35202882263727020787012978441, −4.90217461302165383951306066785, −4.43718168230733038584775330490, −3.96414004902763441188966460948, −3.63685219384694221354548819077, −2.91103777254490309407546115060, −2.75557830537893932966884538548, −2.23419283449759969299113854842, −1.73684536868741167577433724392, −0.992050436900914754116202054484, −0.52597335362696331861488673489, 0.52597335362696331861488673489, 0.992050436900914754116202054484, 1.73684536868741167577433724392, 2.23419283449759969299113854842, 2.75557830537893932966884538548, 2.91103777254490309407546115060, 3.63685219384694221354548819077, 3.96414004902763441188966460948, 4.43718168230733038584775330490, 4.90217461302165383951306066785, 5.35202882263727020787012978441, 5.42755694932790245043144064116, 5.70561057142759781889338646595, 6.23306749822150298562826978274, 7.05327383951325524737312396510, 7.32455680407539074097898174167, 7.46721978945700448306441662723, 7.50760568985551690456235254807, 7.974684030038371362661499655364, 8.332500620763886995163268329317

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