Properties

Label 4-800320-1.1-c1e2-0-3
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·9-s + 6·13-s − 8·17-s − 2·25-s − 2·29-s − 14·37-s − 3·41-s − 2·45-s + 8·49-s + 8·53-s + 7·61-s + 6·65-s + 4·73-s − 5·81-s − 8·85-s + 2·89-s − 14·97-s + 20·101-s + 10·109-s + 8·113-s − 12·117-s − 8·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.447·5-s − 2/3·9-s + 1.66·13-s − 1.94·17-s − 2/5·25-s − 0.371·29-s − 2.30·37-s − 0.468·41-s − 0.298·45-s + 8/7·49-s + 1.09·53-s + 0.896·61-s + 0.744·65-s + 0.468·73-s − 5/9·81-s − 0.867·85-s + 0.211·89-s − 1.42·97-s + 1.99·101-s + 0.957·109-s + 0.752·113-s − 1.10·117-s − 0.727·121-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: \(1\)
Selberg data: \((4,\ 800320,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−8.247658000180988947890253514716, −7.48168001799794547999928985232, −7.11914929389918597980963935584, −6.54019652352669494820665245836, −6.31226569694325206048189619321, −5.79945991911009684576091154250, −5.32394306642124884865393324008, −4.93240178143531949091411594764, −4.11871130856689840336310358761, −3.80130586509208040180634531543, −3.29104930477641380857456310406, −2.42299562930243412872376117621, −2.04867386587910704648619430628, −1.23139974976646853808284811930, 0, 1.23139974976646853808284811930, 2.04867386587910704648619430628, 2.42299562930243412872376117621, 3.29104930477641380857456310406, 3.80130586509208040180634531543, 4.11871130856689840336310358761, 4.93240178143531949091411594764, 5.32394306642124884865393324008, 5.79945991911009684576091154250, 6.31226569694325206048189619321, 6.54019652352669494820665245836, 7.11914929389918597980963935584, 7.48168001799794547999928985232, 8.247658000180988947890253514716

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