Properties

Label 4-1738e2-1.1-c1e2-0-3
Degree $4$
Conductor $3020644$
Sign $-1$
Analytic cond. $192.598$
Root an. cond. $3.72531$
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-s + 2·5-s − 3·9-s + 2·11-s − 2·12-s − 4·15-s + 16-s + 2·20-s − 12·23-s − 7·25-s + 14·27-s + 4·31-s − 4·33-s − 3·36-s − 4·37-s + 2·44-s − 6·45-s + 6·47-s − 2·48-s − 5·49-s + 8·53-s + 4·55-s + 10·59-s − 4·60-s + 64-s + 16·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.894·5-s − 9-s + 0.603·11-s − 0.577·12-s − 1.03·15-s + 1/4·16-s + 0.447·20-s − 2.50·23-s − 7/5·25-s + 2.69·27-s + 0.718·31-s − 0.696·33-s − 1/2·36-s − 0.657·37-s + 0.301·44-s − 0.894·45-s + 0.875·47-s − 0.288·48-s − 5/7·49-s + 1.09·53-s + 0.539·55-s + 1.30·59-s − 0.516·60-s + 1/8·64-s + 1.95·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3020644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3020644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3020644\)    =    \(2^{2} \cdot 11^{2} \cdot 79^{2}\)
Sign: $-1$
Analytic conductor: \(192.598\)
Root analytic conductor: \(3.72531\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3020644,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
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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

−7.46236802123844788529860970104, −6.64493097774474344934567988246, −6.25892605164988854797097226123, −6.11586748167354289065343430906, −5.83005489949556351300916347240, −5.34006059048988750874758197539, −5.14419818902064715285495693173, −4.17874406468249876827955261357, −4.08090613362146692342747948187, −3.30276856154297731238381934791, −2.75823810513610419473379197748, −2.06983106012671431074676520675, −1.88742893280581276183533010694, −0.842556622833874904245322034956, 0, 0.842556622833874904245322034956, 1.88742893280581276183533010694, 2.06983106012671431074676520675, 2.75823810513610419473379197748, 3.30276856154297731238381934791, 4.08090613362146692342747948187, 4.17874406468249876827955261357, 5.14419818902064715285495693173, 5.34006059048988750874758197539, 5.83005489949556351300916347240, 6.11586748167354289065343430906, 6.25892605164988854797097226123, 6.64493097774474344934567988246, 7.46236802123844788529860970104

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