Properties

Label 4-1632e2-1.1-c1e2-0-42
Degree $4$
Conductor $2663424$
Sign $-1$
Analytic cond. $169.822$
Root an. cond. $3.60992$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s + 6·11-s − 2·17-s + 2·19-s − 25-s − 4·27-s − 12·33-s − 6·41-s + 14·43-s + 2·49-s + 4·51-s − 4·57-s − 12·59-s + 8·67-s + 4·73-s + 2·75-s + 5·81-s + 12·83-s − 32·97-s + 18·99-s − 18·107-s − 18·113-s + 5·121-s + 12·123-s + 127-s − 28·129-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 1.80·11-s − 0.485·17-s + 0.458·19-s − 1/5·25-s − 0.769·27-s − 2.08·33-s − 0.937·41-s + 2.13·43-s + 2/7·49-s + 0.560·51-s − 0.529·57-s − 1.56·59-s + 0.977·67-s + 0.468·73-s + 0.230·75-s + 5/9·81-s + 1.31·83-s − 3.24·97-s + 1.80·99-s − 1.74·107-s − 1.69·113-s + 5/11·121-s + 1.08·123-s + 0.0887·127-s − 2.46·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2663424\)    =    \(2^{10} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(169.822\)
Root analytic conductor: \(3.60992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2663424} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2663424,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−7.31936717375093530826366945214, −6.80299500068960285512965295533, −6.56329579539546958275321552423, −6.19889841643513433803233134244, −5.81241206259765754250513501442, −5.21243235840224221221520663568, −5.01602847451418763245834802500, −4.18993695410695019989623309599, −4.10168552133706270535583651662, −3.63699123933177341557225053751, −2.85877051829134286082199945018, −2.22032145426687050578358306238, −1.39440035412653237109928920217, −1.08612370698107395863808052921, 0, 1.08612370698107395863808052921, 1.39440035412653237109928920217, 2.22032145426687050578358306238, 2.85877051829134286082199945018, 3.63699123933177341557225053751, 4.10168552133706270535583651662, 4.18993695410695019989623309599, 5.01602847451418763245834802500, 5.21243235840224221221520663568, 5.81241206259765754250513501442, 6.19889841643513433803233134244, 6.56329579539546958275321552423, 6.80299500068960285512965295533, 7.31936717375093530826366945214

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