Properties

Label 4-760e2-1.1-c1e2-0-14
Degree $4$
Conductor $577600$
Sign $-1$
Analytic cond. $36.8282$
Root an. cond. $2.46345$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 2·5-s + 6·9-s − 8·15-s − 12·17-s + 4·19-s + 3·25-s − 4·27-s + 8·31-s − 12·45-s − 10·49-s − 48·51-s + 16·57-s − 24·59-s + 4·61-s − 4·67-s + 24·71-s + 4·73-s + 12·75-s − 16·79-s − 37·81-s + 24·85-s + 32·93-s − 8·95-s + 12·101-s − 28·103-s + 12·107-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 2·9-s − 2.06·15-s − 2.91·17-s + 0.917·19-s + 3/5·25-s − 0.769·27-s + 1.43·31-s − 1.78·45-s − 1.42·49-s − 6.72·51-s + 2.11·57-s − 3.12·59-s + 0.512·61-s − 0.488·67-s + 2.84·71-s + 0.468·73-s + 1.38·75-s − 1.80·79-s − 4.11·81-s + 2.60·85-s + 3.31·93-s − 0.820·95-s + 1.19·101-s − 2.75·103-s + 1.16·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 577600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(577600\)    =    \(2^{6} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(36.8282\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 577600,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
19$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.347343430009299439276351390943, −7.71273110823499542846308181544, −7.70342498742006188501994585256, −7.06215723111114144957068885167, −6.39347529836665585887623697697, −6.27087624192875571051851265258, −5.14081916011398824138614106106, −4.74913960233451073492131590460, −4.12250433368686324236236368171, −3.82183399469630822167941819328, −3.12234796120650330097153328391, −2.76929890617261215013507568311, −2.31822883421800731675688556606, −1.57301709011122909909496546092, 0, 1.57301709011122909909496546092, 2.31822883421800731675688556606, 2.76929890617261215013507568311, 3.12234796120650330097153328391, 3.82183399469630822167941819328, 4.12250433368686324236236368171, 4.74913960233451073492131590460, 5.14081916011398824138614106106, 6.27087624192875571051851265258, 6.39347529836665585887623697697, 7.06215723111114144957068885167, 7.70342498742006188501994585256, 7.71273110823499542846308181544, 8.347343430009299439276351390943

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