Properties

Label 4-2550e2-1.1-c1e2-0-17
Degree $4$
Conductor $6502500$
Sign $1$
Analytic cond. $414.605$
Root an. cond. $4.51241$
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  − 4-s − 9-s − 2·11-s + 16-s + 14·19-s + 20·29-s − 10·31-s + 36-s − 20·41-s + 2·44-s − 11·49-s + 4·61-s − 64-s − 20·71-s − 14·76-s − 6·79-s + 81-s − 8·89-s + 2·99-s − 14·101-s − 38·109-s − 20·116-s − 19·121-s + 10·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s + 3.21·19-s + 3.71·29-s − 1.79·31-s + 1/6·36-s − 3.12·41-s + 0.301·44-s − 1.57·49-s + 0.512·61-s − 1/8·64-s − 2.37·71-s − 1.60·76-s − 0.675·79-s + 1/9·81-s − 0.847·89-s + 0.201·99-s − 1.39·101-s − 3.63·109-s − 1.85·116-s − 1.72·121-s + 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6502500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(414.605\)
Root analytic conductor: \(4.51241\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6502500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.507938705\)
\(L(\frac12)\) \(\approx\) \(1.507938705\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
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

−9.151068943723225360059724254827, −8.676635831766271897420256231594, −8.261827334836010057403941007254, −7.975628870477223772557022726042, −7.76955741657710922557876033221, −7.08060251782171112510088437948, −6.71030710522872342213364328218, −6.70511439410044147750511525501, −5.74990505732547804528468477413, −5.51988700155461781919982185143, −5.18291280135879573308046411090, −4.89070280874311717662520231610, −4.42717894193652637504258172345, −3.81676111435583416335504296087, −3.14482496881194778708854965669, −3.07641607559140570256451621163, −2.70291583363309397495499174844, −1.54279479822898448616219573379, −1.36590696849494497851558601108, −0.43509260223608025489311448249, 0.43509260223608025489311448249, 1.36590696849494497851558601108, 1.54279479822898448616219573379, 2.70291583363309397495499174844, 3.07641607559140570256451621163, 3.14482496881194778708854965669, 3.81676111435583416335504296087, 4.42717894193652637504258172345, 4.89070280874311717662520231610, 5.18291280135879573308046411090, 5.51988700155461781919982185143, 5.74990505732547804528468477413, 6.70511439410044147750511525501, 6.71030710522872342213364328218, 7.08060251782171112510088437948, 7.76955741657710922557876033221, 7.975628870477223772557022726042, 8.261827334836010057403941007254, 8.676635831766271897420256231594, 9.151068943723225360059724254827

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