Properties

Label 4-800320-1.1-c1e2-0-1
Degree $4$
Conductor $800320$
Sign $1$
Analytic cond. $51.0290$
Root an. cond. $2.67272$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 5·9-s + 12·13-s + 5·17-s − 2·25-s + 4·29-s − 21·37-s − 41-s + 5·45-s + 10·49-s + 53-s + 2·61-s + 12·65-s − 4·73-s + 16·81-s + 5·85-s − 16·89-s − 7·97-s − 28·101-s + 7·109-s + 11·113-s + 60·117-s − 5·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.447·5-s + 5/3·9-s + 3.32·13-s + 1.21·17-s − 2/5·25-s + 0.742·29-s − 3.45·37-s − 0.156·41-s + 0.745·45-s + 10/7·49-s + 0.137·53-s + 0.256·61-s + 1.48·65-s − 0.468·73-s + 16/9·81-s + 0.542·85-s − 1.69·89-s − 0.710·97-s − 2.78·101-s + 0.670·109-s + 1.03·113-s + 5.54·117-s − 0.454·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(800320\)    =    \(2^{6} \cdot 5 \cdot 41 \cdot 61\)
Sign: $1$
Analytic conductor: \(51.0290\)
Root analytic conductor: \(2.67272\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{800320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 800320,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.476411754\)
\(L(\frac12)\) \(\approx\) \(3.476411754\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 83 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 112 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.288559563281819420350494292332, −7.942313961823255569318744323020, −7.17887758609772624441644062622, −6.88427827452403242273004071673, −6.58018193046949069737628023531, −5.91441632932643969745087073460, −5.59710797076793673100512138178, −5.23349678651743309954780579447, −4.28108882289940864842916454478, −4.07172866143636139222097398660, −3.45863745802712588196399639346, −3.20707633685834028559904709709, −1.99302602612246717545843835322, −1.40628897199210914480781914950, −1.11523311835702496158253562486, 1.11523311835702496158253562486, 1.40628897199210914480781914950, 1.99302602612246717545843835322, 3.20707633685834028559904709709, 3.45863745802712588196399639346, 4.07172866143636139222097398660, 4.28108882289940864842916454478, 5.23349678651743309954780579447, 5.59710797076793673100512138178, 5.91441632932643969745087073460, 6.58018193046949069737628023531, 6.88427827452403242273004071673, 7.17887758609772624441644062622, 7.942313961823255569318744323020, 8.288559563281819420350494292332

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