Properties

Label 4-3066624-1.1-c1e2-0-6
Degree $4$
Conductor $3066624$
Sign $1$
Analytic cond. $195.530$
Root an. cond. $3.73941$
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  − 2·3-s − 4·5-s + 3·9-s + 11-s + 8·15-s + 2·25-s − 4·27-s − 2·33-s + 12·37-s − 12·45-s + 2·49-s + 12·53-s − 4·55-s − 8·59-s + 24·67-s + 16·71-s − 4·75-s + 5·81-s − 12·89-s + 4·97-s + 3·99-s − 24·111-s + 20·113-s + 121-s + 28·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 9-s + 0.301·11-s + 2.06·15-s + 2/5·25-s − 0.769·27-s − 0.348·33-s + 1.97·37-s − 1.78·45-s + 2/7·49-s + 1.64·53-s − 0.539·55-s − 1.04·59-s + 2.93·67-s + 1.89·71-s − 0.461·75-s + 5/9·81-s − 1.27·89-s + 0.406·97-s + 0.301·99-s − 2.27·111-s + 1.88·113-s + 1/11·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3066624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3066624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3066624\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(195.530\)
Root analytic conductor: \(3.73941\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3066624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8595943237\)
\(L(\frac12)\) \(\approx\) \(0.8595943237\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−7.35824269240573348608620103444, −7.19819071506851382078022350491, −6.84172401965691206022768509346, −6.19015485249586140740060842128, −5.96726719550806915467982642379, −5.46405190258323470080919041481, −4.94084641236370924155494583195, −4.49914350038850325136622917050, −4.12248457413213219306418931774, −3.76305426337925729255035834153, −3.38108759817336471336235218518, −2.54287436292890018722810885401, −1.94546966374609697836921373063, −0.918992040659386671383695016650, −0.50708313336612890338490237295, 0.50708313336612890338490237295, 0.918992040659386671383695016650, 1.94546966374609697836921373063, 2.54287436292890018722810885401, 3.38108759817336471336235218518, 3.76305426337925729255035834153, 4.12248457413213219306418931774, 4.49914350038850325136622917050, 4.94084641236370924155494583195, 5.46405190258323470080919041481, 5.96726719550806915467982642379, 6.19015485249586140740060842128, 6.84172401965691206022768509346, 7.19819071506851382078022350491, 7.35824269240573348608620103444

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