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Properties

Label 4-2960e2-1.1-c0e2-0-5
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  + 2·3-s + 2·5-s + 9-s + 4·15-s + 3·25-s − 2·27-s − 2·31-s + 2·43-s + 2·45-s + 49-s − 2·61-s + 2·71-s + 2·73-s + 6·75-s − 2·79-s − 4·81-s − 2·89-s − 4·93-s + 2·97-s + 2·103-s − 2·109-s − 2·113-s + 121-s + 4·125-s + 127-s + 4·129-s + 131-s + ⋯
L(s)  = 1  + 2·3-s + 2·5-s + 9-s + 4·15-s + 3·25-s − 2·27-s − 2·31-s + 2·43-s + 2·45-s + 49-s − 2·61-s + 2·71-s + 2·73-s + 6·75-s − 2·79-s − 4·81-s − 2·89-s − 4·93-s + 2·97-s + 2·103-s − 2·109-s − 2·113-s + 121-s + 4·125-s + 127-s + 4·129-s + 131-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

\(L(\frac{1}{2})\) \(\approx\) \(3.890404117\)
\(L(\frac12)\) \(\approx\) \(3.890404117\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
47$C_2^2$ \( 1 - T^{2} + T^{4} \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−9.059475940381516885011002168660, −8.885091469160652560019143267477, −8.554746379815246561482071674591, −8.103330691581431595383593909047, −7.55514317719899712486690314678, −7.50542736051155391613598042313, −6.89223291455868412644307205074, −6.50641991675492152299541243899, −5.92413727805773491659101914419, −5.72944983149797268441627081549, −5.41707648081504846016104737054, −4.95897654898646017536991845606, −4.24333617522462865527958488157, −3.90586087607781531562269818988, −3.16811921201409471414787485269, −3.14461770674360719510883973381, −2.33499886408065518786963911309, −2.31855443981185852353337202802, −1.83618408207245179456766491988, −1.16579340363449664436353293054, 1.16579340363449664436353293054, 1.83618408207245179456766491988, 2.31855443981185852353337202802, 2.33499886408065518786963911309, 3.14461770674360719510883973381, 3.16811921201409471414787485269, 3.90586087607781531562269818988, 4.24333617522462865527958488157, 4.95897654898646017536991845606, 5.41707648081504846016104737054, 5.72944983149797268441627081549, 5.92413727805773491659101914419, 6.50641991675492152299541243899, 6.89223291455868412644307205074, 7.50542736051155391613598042313, 7.55514317719899712486690314678, 8.103330691581431595383593909047, 8.554746379815246561482071674591, 8.885091469160652560019143267477, 9.059475940381516885011002168660

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