Properties

Label 4-1620e2-1.1-c3e2-0-0
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $9136.12$
Root an. cond. $9.77666$
Motivic weight $3$
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  − 5·5-s − 2·7-s − 30·11-s + 4·13-s + 180·17-s − 56·19-s − 120·23-s − 210·29-s + 4·31-s + 10·35-s + 400·37-s − 240·41-s + 136·43-s + 120·47-s + 343·49-s − 60·53-s + 150·55-s + 450·59-s + 166·61-s − 20·65-s − 908·67-s − 2.04e3·71-s − 500·73-s + 60·77-s + 916·79-s + 1.14e3·83-s − 900·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.107·7-s − 0.822·11-s + 0.0853·13-s + 2.56·17-s − 0.676·19-s − 1.08·23-s − 1.34·29-s + 0.0231·31-s + 0.0482·35-s + 1.77·37-s − 0.914·41-s + 0.482·43-s + 0.372·47-s + 49-s − 0.155·53-s + 0.367·55-s + 0.992·59-s + 0.348·61-s − 0.0381·65-s − 1.65·67-s − 3.40·71-s − 0.801·73-s + 0.0888·77-s + 1.30·79-s + 1.50·83-s − 1.14·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(9136.12\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1109970691\)
\(L(\frac12)\) \(\approx\) \(0.1109970691\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + p T + p^{2} T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 339 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 4 T - 2181 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 - 90 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 28 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 120 T + 2233 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 210 T + 19711 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 4 T - 29775 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 200 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 240 T - 11321 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 136 T - 61011 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 120 T - 89423 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 30 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 450 T - 2879 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 166 T - 199425 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 908 T + 523701 T^{2} + 908 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 1020 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 250 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 916 T + 346017 T^{2} - 916 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 1140 T + 727813 T^{2} - 1140 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 420 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 1538 T + 1452771 T^{2} + 1538 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.577975142715556788500628192157, −8.713671906088203592311723777997, −8.368239522967900127809999200971, −7.86927649750114272249377465992, −7.80947908342359151375510984630, −7.25866805029369186443985096867, −7.09163031409123613086511297252, −6.17528960484236701451022299210, −5.99225580246452438874083777557, −5.52172674261808539951620271367, −5.34890602703545337268806780692, −4.52610248911436309000694764837, −4.24115095677107864646307847472, −3.60785198718358548573447097410, −3.42904024109421979011817073283, −2.53295497382072620116038362480, −2.48946374736915228736623506539, −1.37232588779270444710161745246, −1.17473545460130699301810812103, −0.07681495809115050831755858804, 0.07681495809115050831755858804, 1.17473545460130699301810812103, 1.37232588779270444710161745246, 2.48946374736915228736623506539, 2.53295497382072620116038362480, 3.42904024109421979011817073283, 3.60785198718358548573447097410, 4.24115095677107864646307847472, 4.52610248911436309000694764837, 5.34890602703545337268806780692, 5.52172674261808539951620271367, 5.99225580246452438874083777557, 6.17528960484236701451022299210, 7.09163031409123613086511297252, 7.25866805029369186443985096867, 7.80947908342359151375510984630, 7.86927649750114272249377465992, 8.368239522967900127809999200971, 8.713671906088203592311723777997, 9.577975142715556788500628192157

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