Properties

Label 4-332928-1.1-c1e2-0-31
Degree $4$
Conductor $332928$
Sign $-1$
Analytic cond. $21.2277$
Root an. cond. $2.14647$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 3·5-s − 6-s − 8-s − 2·9-s − 3·10-s + 12-s + 3·15-s + 16-s + 2·18-s − 2·19-s + 3·20-s − 6·23-s − 24-s − 25-s − 5·27-s − 12·29-s − 3·30-s − 32-s − 2·36-s + 2·38-s − 3·40-s + 7·43-s − 6·45-s + 6·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.288·12-s + 0.774·15-s + 1/4·16-s + 0.471·18-s − 0.458·19-s + 0.670·20-s − 1.25·23-s − 0.204·24-s − 1/5·25-s − 0.962·27-s − 2.22·29-s − 0.547·30-s − 0.176·32-s − 1/3·36-s + 0.324·38-s − 0.474·40-s + 1.06·43-s − 0.894·45-s + 0.884·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(332928\)    =    \(2^{7} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(21.2277\)
Root analytic conductor: \(2.14647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 332928,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3$C_2$ \( 1 - T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−8.651707926225404781781052275806, −7.981825706181436972776941879022, −7.86045522688470142906013735097, −7.28949864139311801546943708983, −6.61090493813459854147139575588, −6.08251093950055890212735051070, −5.82024171278327378828822460508, −5.44491032950393586920797555705, −4.64782609107135243902649102294, −3.83578581819055462071722812434, −3.40143729930981129634384981430, −2.52882026207307242437407028298, −2.06227410289181150586814080815, −1.62901038406687568185313451564, 0, 1.62901038406687568185313451564, 2.06227410289181150586814080815, 2.52882026207307242437407028298, 3.40143729930981129634384981430, 3.83578581819055462071722812434, 4.64782609107135243902649102294, 5.44491032950393586920797555705, 5.82024171278327378828822460508, 6.08251093950055890212735051070, 6.61090493813459854147139575588, 7.28949864139311801546943708983, 7.86045522688470142906013735097, 7.981825706181436972776941879022, 8.651707926225404781781052275806

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