Properties

Label 4-1848e2-1.1-c1e2-0-19
Degree $4$
Conductor $3415104$
Sign $-1$
Analytic cond. $217.749$
Root an. cond. $3.84140$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·5-s + 3·9-s + 8·15-s − 8·23-s + 2·25-s + 4·27-s − 16·31-s − 20·37-s + 12·45-s − 16·47-s + 49-s + 12·53-s + 8·59-s + 24·67-s − 16·69-s + 8·71-s + 4·75-s + 5·81-s + 12·89-s − 32·93-s + 20·97-s − 40·111-s − 28·113-s − 32·115-s − 11·121-s − 28·125-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s + 2.06·15-s − 1.66·23-s + 2/5·25-s + 0.769·27-s − 2.87·31-s − 3.28·37-s + 1.78·45-s − 2.33·47-s + 1/7·49-s + 1.64·53-s + 1.04·59-s + 2.93·67-s − 1.92·69-s + 0.949·71-s + 0.461·75-s + 5/9·81-s + 1.27·89-s − 3.31·93-s + 2.03·97-s − 3.79·111-s − 2.63·113-s − 2.98·115-s − 121-s − 2.50·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3415104\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(217.749\)
Root analytic conductor: \(3.84140\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3415104} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3415104,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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

−7.35653811838304932393623429283, −6.86573975539025229905898029503, −6.42136658293515809263418102895, −6.18287900832404807661428217465, −5.41807067094670761229863645645, −5.25019213638990204722565722077, −5.07252108538521589319110157600, −3.90820666302401882520500245752, −3.69478854082878580470787829226, −3.56921408081238054859871707771, −2.55256266755253863914783357450, −2.06515692945354055956624655608, −1.95860198383842967586361157419, −1.43997509206912042814006725043, 0, 1.43997509206912042814006725043, 1.95860198383842967586361157419, 2.06515692945354055956624655608, 2.55256266755253863914783357450, 3.56921408081238054859871707771, 3.69478854082878580470787829226, 3.90820666302401882520500245752, 5.07252108538521589319110157600, 5.25019213638990204722565722077, 5.41807067094670761229863645645, 6.18287900832404807661428217465, 6.42136658293515809263418102895, 6.86573975539025229905898029503, 7.35653811838304932393623429283

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