Properties

Label 4-2475e2-1.1-c1e2-0-15
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $390.575$
Root an. cond. $4.44555$
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  + 4·4-s + 2·11-s + 12·16-s + 14·19-s − 12·29-s − 14·31-s + 12·41-s + 8·44-s + 13·49-s + 10·61-s + 32·64-s + 24·71-s + 56·76-s + 8·79-s + 12·89-s − 22·109-s − 48·116-s + 3·121-s − 56·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2·4-s + 0.603·11-s + 3·16-s + 3.21·19-s − 2.22·29-s − 2.51·31-s + 1.87·41-s + 1.20·44-s + 13/7·49-s + 1.28·61-s + 4·64-s + 2.84·71-s + 6.42·76-s + 0.900·79-s + 1.27·89-s − 2.10·109-s − 4.45·116-s + 3/11·121-s − 5.02·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(390.575\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6125625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.074195548\)
\(L(\frac12)\) \(\approx\) \(6.074195548\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
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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

−9.106151442260302025402979844486, −9.034780048850486020261776817385, −7.931761786785549635006573289056, −7.87355519110775295375839220454, −7.48448661693733516914137586594, −7.32945710960031162431914299841, −6.76532272327669584285943252451, −6.69590727417759405042490080706, −5.87865026222168586660695894397, −5.67037405981371547104126721338, −5.30096916079741406154216323972, −5.21455883905553777613422073597, −3.88467657385001751869006516793, −3.85669466932834557595154315641, −3.40712924817309745395910884968, −2.93729826510640535921209487457, −2.21591638614725870500589481145, −2.07194957795591154620869899081, −1.25845262176396434545503229591, −0.894032717299712361175198817605, 0.894032717299712361175198817605, 1.25845262176396434545503229591, 2.07194957795591154620869899081, 2.21591638614725870500589481145, 2.93729826510640535921209487457, 3.40712924817309745395910884968, 3.85669466932834557595154315641, 3.88467657385001751869006516793, 5.21455883905553777613422073597, 5.30096916079741406154216323972, 5.67037405981371547104126721338, 5.87865026222168586660695894397, 6.69590727417759405042490080706, 6.76532272327669584285943252451, 7.32945710960031162431914299841, 7.48448661693733516914137586594, 7.87355519110775295375839220454, 7.931761786785549635006573289056, 9.034780048850486020261776817385, 9.106151442260302025402979844486

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