Properties

Label 4-800384-1.1-c1e2-0-1
Degree $4$
Conductor $800384$
Sign $-1$
Analytic cond. $51.0331$
Root an. cond. $2.67277$
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  − 4·5-s − 2·9-s + 2·13-s + 6·25-s − 2·29-s − 5·37-s − 2·41-s + 8·45-s + 4·49-s + 14·53-s − 2·61-s − 8·65-s + 8·73-s − 5·81-s + 16·89-s + 6·97-s + 6·101-s − 8·109-s − 2·113-s − 4·117-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯
L(s)  = 1  − 1.78·5-s − 2/3·9-s + 0.554·13-s + 6/5·25-s − 0.371·29-s − 0.821·37-s − 0.312·41-s + 1.19·45-s + 4/7·49-s + 1.92·53-s − 0.256·61-s − 0.992·65-s + 0.936·73-s − 5/9·81-s + 1.69·89-s + 0.609·97-s + 0.597·101-s − 0.766·109-s − 0.188·113-s − 0.369·117-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(800384\)    =    \(2^{7} \cdot 13^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(51.0331\)
Root analytic conductor: \(2.67277\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 800384,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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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

−7.953935566717030733901991575500, −7.65694216779119187181199716327, −7.27585552886302214491127407523, −6.75144409225339197990099963073, −6.32059922724817283548589764147, −5.69730363612505811610472063621, −5.28333946520972550880688033014, −4.75081578367799662546097597178, −4.07807451717637017208172802748, −3.79565018233219380100043453277, −3.38117068008461867828720146136, −2.73162985827577375517490103516, −1.98607535979107465010264954773, −0.912357969252171054784897597682, 0, 0.912357969252171054784897597682, 1.98607535979107465010264954773, 2.73162985827577375517490103516, 3.38117068008461867828720146136, 3.79565018233219380100043453277, 4.07807451717637017208172802748, 4.75081578367799662546097597178, 5.28333946520972550880688033014, 5.69730363612505811610472063621, 6.32059922724817283548589764147, 6.75144409225339197990099963073, 7.27585552886302214491127407523, 7.65694216779119187181199716327, 7.953935566717030733901991575500

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