Properties

Label 4-2960e2-1.1-c0e2-0-4
Degree $4$
Conductor $8761600$
Sign $1$
Analytic cond. $2.18221$
Root an. cond. $1.21541$
Motivic weight $0$
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  + 2·5-s + 3·25-s + 2·29-s + 2·37-s − 2·53-s + 2·61-s − 2·73-s − 81-s − 2·89-s + 4·97-s − 2·109-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2·5-s + 3·25-s + 2·29-s + 2·37-s − 2·53-s + 2·61-s − 2·73-s − 81-s − 2·89-s + 4·97-s − 2·109-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8761600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8761600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8761600\)    =    \(2^{8} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(2.18221\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8761600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.313013518\)
\(L(\frac12)\) \(\approx\) \(2.313013518\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
31$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_1$ \( ( 1 - T )^{4} \)
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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

−8.981561352064478990295532479198, −8.913067598529495488702094163069, −8.496464401155093086501324871858, −7.995737472845140081692412820653, −7.67040064438679314514119852193, −7.14994014115242988470709014104, −6.61266789467305997149464159372, −6.47376079546832560673368628954, −6.11218130779767829218663894678, −5.78028760957398732186724775059, −5.20662776530580697136081023284, −5.06694596616062340701323294831, −4.37015151238388350983553211857, −4.29669721301503063957824826984, −3.32047058723932312643792262325, −2.96644912324728096945516213202, −2.45970070691212981498848123866, −2.23163055592869945829565136046, −1.31579764839845224557079622792, −1.14820562126073964932029727565, 1.14820562126073964932029727565, 1.31579764839845224557079622792, 2.23163055592869945829565136046, 2.45970070691212981498848123866, 2.96644912324728096945516213202, 3.32047058723932312643792262325, 4.29669721301503063957824826984, 4.37015151238388350983553211857, 5.06694596616062340701323294831, 5.20662776530580697136081023284, 5.78028760957398732186724775059, 6.11218130779767829218663894678, 6.47376079546832560673368628954, 6.61266789467305997149464159372, 7.14994014115242988470709014104, 7.67040064438679314514119852193, 7.995737472845140081692412820653, 8.496464401155093086501324871858, 8.913067598529495488702094163069, 8.981561352064478990295532479198

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