Properties

Label 4-80e4-1.1-c1e2-0-15
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
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  − 4·9-s + 12·13-s + 12·17-s + 12·37-s + 4·49-s + 12·53-s − 12·61-s − 4·73-s + 7·81-s + 12·89-s + 20·97-s + 24·101-s − 36·109-s − 12·113-s − 48·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s + 163-s + 167-s + 82·169-s + ⋯
L(s)  = 1  − 4/3·9-s + 3.32·13-s + 2.91·17-s + 1.97·37-s + 4/7·49-s + 1.64·53-s − 1.53·61-s − 0.468·73-s + 7/9·81-s + 1.27·89-s + 2.03·97-s + 2.38·101-s − 3.44·109-s − 1.12·113-s − 4.43·117-s − 1.27·121-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 − 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.571312973\)
\(L(\frac12)\) \(\approx\) \(4.571312973\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.087131187909989571183711607087, −8.018244258768805602857127926165, −7.52439444935549018748440260285, −7.38682329869242195978431875858, −6.54902116583160078457498181296, −6.32977822392414254727262179480, −5.97686817344060701068800430240, −5.84099598424444258788863174574, −5.50433365104009204150114804745, −5.18002230695136299284868042216, −4.58137625351535853807951665470, −3.98470810532932297185752895973, −3.73588193570788457371725795540, −3.47530558600492980082378826755, −2.98224415861376352629465913655, −2.79489532307506723284643691065, −2.03602017409403839193217127194, −1.32971374542087072473266045345, −1.07295249426730664996360014568, −0.65632583199524671560069537478, 0.65632583199524671560069537478, 1.07295249426730664996360014568, 1.32971374542087072473266045345, 2.03602017409403839193217127194, 2.79489532307506723284643691065, 2.98224415861376352629465913655, 3.47530558600492980082378826755, 3.73588193570788457371725795540, 3.98470810532932297185752895973, 4.58137625351535853807951665470, 5.18002230695136299284868042216, 5.50433365104009204150114804745, 5.84099598424444258788863174574, 5.97686817344060701068800430240, 6.32977822392414254727262179480, 6.54902116583160078457498181296, 7.38682329869242195978431875858, 7.52439444935549018748440260285, 8.018244258768805602857127926165, 8.087131187909989571183711607087

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