Properties

Label 4-280e2-1.1-c1e2-0-13
Degree $4$
Conductor $78400$
Sign $-1$
Analytic cond. $4.99885$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 3·9-s − 2·11-s − 4·16-s + 6·18-s + 4·22-s + 25-s + 8·32-s − 6·36-s + 12·43-s − 4·44-s − 7·49-s − 2·50-s − 8·64-s − 28·67-s − 24·86-s + 14·98-s + 6·99-s + 2·100-s − 4·107-s − 16·113-s − 19·121-s + 127-s + 131-s + 56·134-s + 137-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 9-s − 0.603·11-s − 16-s + 1.41·18-s + 0.852·22-s + 1/5·25-s + 1.41·32-s − 36-s + 1.82·43-s − 0.603·44-s − 49-s − 0.282·50-s − 64-s − 3.42·67-s − 2.58·86-s + 1.41·98-s + 0.603·99-s + 1/5·100-s − 0.386·107-s − 1.50·113-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 4.83·134-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(4.99885\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 78400,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.413833215018796083736788450286, −8.905620132103282237099470021544, −8.690318352415095273348099800880, −8.032256305048544541115493926269, −7.63572025887558713136221609863, −7.27326475782480216593475577334, −6.46570458456308613518881961491, −6.02378822140684312049628434559, −5.35555342291496416407146876793, −4.72232396451229340241889116709, −3.99977094286265568450903155609, −2.96099201703757334117530089963, −2.47007791992779484158478737131, −1.37226864740468197156839338898, 0, 1.37226864740468197156839338898, 2.47007791992779484158478737131, 2.96099201703757334117530089963, 3.99977094286265568450903155609, 4.72232396451229340241889116709, 5.35555342291496416407146876793, 6.02378822140684312049628434559, 6.46570458456308613518881961491, 7.27326475782480216593475577334, 7.63572025887558713136221609863, 8.032256305048544541115493926269, 8.690318352415095273348099800880, 8.905620132103282237099470021544, 9.413833215018796083736788450286

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