Properties

Label 4-10800-1.1-c1e2-0-0
Degree $4$
Conductor $10800$
Sign $1$
Analytic cond. $0.688617$
Root an. cond. $0.910949$
Motivic weight $1$
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-s + 3-s − 4-s + 6-s − 3·8-s + 9-s + 8·11-s − 12-s − 4·13-s − 16-s + 18-s + 8·22-s − 3·24-s + 25-s − 4·26-s + 27-s + 5·32-s + 8·33-s − 36-s − 20·37-s − 4·39-s − 8·44-s − 16·47-s − 48-s − 14·49-s + 50-s + 4·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 2.41·11-s − 0.288·12-s − 1.10·13-s − 1/4·16-s + 0.235·18-s + 1.70·22-s − 0.612·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.883·32-s + 1.39·33-s − 1/6·36-s − 3.28·37-s − 0.640·39-s − 1.20·44-s − 2.33·47-s − 0.144·48-s − 2·49-s + 0.141·50-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10800\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.688617\)
Root analytic conductor: \(0.910949\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10800,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.471082268\)
\(L(\frac12)\) \(\approx\) \(1.471082268\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−11.57318950916085697381955699128, −11.16070118695890574879383709815, −10.00510556501914029619044642425, −9.738851148469479131038939478590, −9.265565204604989643425731491439, −8.554588868470699894187777077269, −8.330811446905088069540074876671, −7.12360478295630625617848815878, −6.76955885158912340897756950945, −6.13587545605028508104380795249, −5.04860090093668311608273372623, −4.68831052899409581492283307982, −3.63412818236731287070311550143, −3.39274372810076015084631955767, −1.82346513010869405208007428430, 1.82346513010869405208007428430, 3.39274372810076015084631955767, 3.63412818236731287070311550143, 4.68831052899409581492283307982, 5.04860090093668311608273372623, 6.13587545605028508104380795249, 6.76955885158912340897756950945, 7.12360478295630625617848815878, 8.330811446905088069540074876671, 8.554588868470699894187777077269, 9.265565204604989643425731491439, 9.738851148469479131038939478590, 10.00510556501914029619044642425, 11.16070118695890574879383709815, 11.57318950916085697381955699128

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