Properties

Label 4-4600e2-1.1-c1e2-0-0
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 10·11-s − 14·19-s − 10·29-s + 4·31-s + 22·41-s + 13·49-s + 28·59-s + 20·61-s − 20·71-s − 14·79-s − 5·81-s − 20·89-s − 20·99-s + 36·101-s − 4·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + ⋯
L(s)  = 1  + 2/3·9-s − 3.01·11-s − 3.21·19-s − 1.85·29-s + 0.718·31-s + 3.43·41-s + 13/7·49-s + 3.64·59-s + 2.56·61-s − 2.37·71-s − 1.57·79-s − 5/9·81-s − 2.11·89-s − 2.01·99-s + 3.58·101-s − 0.383·109-s + 4.81·121-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 + 1.92·169-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9709599998\)
\(L(\frac12)\) \(\approx\) \(0.9709599998\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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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

−8.424787667637716630155467492977, −8.264269131570749903758974397380, −7.59778837321851120194824909342, −7.55609953443471149342391748458, −7.09066014002348235488003929050, −6.89796241843628698252813327936, −6.20050033256577282849210668107, −5.73041631065807757287145543299, −5.70166997028613125475032034557, −5.34843722596875682120949105119, −4.57437683728827654822942559735, −4.51207667870896595434632516649, −3.92718986990685228776643370656, −3.86423869656204061429269847821, −2.83057015664671695292213197785, −2.58279445913772675595450311801, −2.16377438635815451101202007456, −2.07690155642178728129418253807, −0.927615109624252587194619249141, −0.31162210229232758839415089723, 0.31162210229232758839415089723, 0.927615109624252587194619249141, 2.07690155642178728129418253807, 2.16377438635815451101202007456, 2.58279445913772675595450311801, 2.83057015664671695292213197785, 3.86423869656204061429269847821, 3.92718986990685228776643370656, 4.51207667870896595434632516649, 4.57437683728827654822942559735, 5.34843722596875682120949105119, 5.70166997028613125475032034557, 5.73041631065807757287145543299, 6.20050033256577282849210668107, 6.89796241843628698252813327936, 7.09066014002348235488003929050, 7.55609953443471149342391748458, 7.59778837321851120194824909342, 8.264269131570749903758974397380, 8.424787667637716630155467492977

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