Properties

Label 4-826875-1.1-c1e2-0-19
Degree $4$
Conductor $826875$
Sign $-1$
Analytic cond. $52.7222$
Root an. cond. $2.69462$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s + 9-s − 3·12-s + 5·16-s − 4·17-s + 27-s − 3·36-s + 20·37-s + 20·41-s − 8·43-s − 16·47-s + 5·48-s − 7·49-s − 4·51-s − 8·59-s − 3·64-s − 24·67-s + 12·68-s + 81-s − 24·83-s − 12·89-s + 12·101-s − 3·108-s + 28·109-s + 20·111-s − 6·121-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s + 1/3·9-s − 0.866·12-s + 5/4·16-s − 0.970·17-s + 0.192·27-s − 1/2·36-s + 3.28·37-s + 3.12·41-s − 1.21·43-s − 2.33·47-s + 0.721·48-s − 49-s − 0.560·51-s − 1.04·59-s − 3/8·64-s − 2.93·67-s + 1.45·68-s + 1/9·81-s − 2.63·83-s − 1.27·89-s + 1.19·101-s − 0.288·108-s + 2.68·109-s + 1.89·111-s − 0.545·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 826875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(826875\)    =    \(3^{3} \cdot 5^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(52.7222\)
Root analytic conductor: \(2.69462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{826875} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 826875,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$ \( 1 - T \)
5 \( 1 \)
7$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.023793216333534711017938369158, −7.68003844385765985083943135134, −7.39848909917701976603786239429, −6.57558779208505392879804927456, −6.12621854048128780982239967429, −5.86497653842965752491328535183, −5.05351399151288949834601255152, −4.52160444884874669934496061646, −4.40145701518461031613845247878, −3.98361923014175761679111177485, −3.01549784140686353642765162463, −2.88208540738665705106266854904, −1.89845619892689055490680198063, −1.06721576698113432176731915153, 0, 1.06721576698113432176731915153, 1.89845619892689055490680198063, 2.88208540738665705106266854904, 3.01549784140686353642765162463, 3.98361923014175761679111177485, 4.40145701518461031613845247878, 4.52160444884874669934496061646, 5.05351399151288949834601255152, 5.86497653842965752491328535183, 6.12621854048128780982239967429, 6.57558779208505392879804927456, 7.39848909917701976603786239429, 7.68003844385765985083943135134, 8.023793216333534711017938369158

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