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Properties

Label 4-91e2-1.1-c1e2-0-0
Degree $4$
Conductor $8281$
Sign $1$
Analytic cond. $0.528003$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·4-s + 7-s − 5·8-s + 3·9-s + 3·11-s − 2·13-s − 14-s + 5·16-s − 7·17-s − 3·18-s + 7·19-s − 3·22-s + 6·23-s + 5·25-s + 2·26-s + 2·28-s − 10·29-s − 10·32-s + 7·34-s + 6·36-s − 8·37-s − 7·38-s + 4·43-s + 6·44-s − 6·46-s − 7·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s + 0.377·7-s − 1.76·8-s + 9-s + 0.904·11-s − 0.554·13-s − 0.267·14-s + 5/4·16-s − 1.69·17-s − 0.707·18-s + 1.60·19-s − 0.639·22-s + 1.25·23-s + 25-s + 0.392·26-s + 0.377·28-s − 1.85·29-s − 1.76·32-s + 1.20·34-s + 36-s − 1.31·37-s − 1.13·38-s + 0.609·43-s + 0.904·44-s − 0.884·46-s − 1.02·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.528003\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8281,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7980576986\)
\(L(\frac12)\) \(\approx\) \(0.7980576986\)
\(L(\frac{3}{2})\) 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)$
bad7$C_2$ \( 1 - T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−14.80902644681530851113949964493, −13.87282216963460635583314164429, −13.10547305650341557205541427777, −12.75164762652578582436312757752, −12.10984816238046023892656484342, −11.39858744034346832621337418871, −11.37304074217237000786930589773, −10.65772235316585565648228861879, −9.849000502889582452239528035307, −9.353677272744523833793081283445, −8.991797058531261839995299601227, −8.406693231176218749553958013660, −7.30209499879233676382938122844, −7.06337995585277557712301026186, −6.63224762833152679414714385275, −5.62600571695683110288567633757, −4.89464592389451657134899984228, −3.80222178285826781011550157876, −2.83477923611271154726898378845, −1.63605626214557679743548316152, 1.63605626214557679743548316152, 2.83477923611271154726898378845, 3.80222178285826781011550157876, 4.89464592389451657134899984228, 5.62600571695683110288567633757, 6.63224762833152679414714385275, 7.06337995585277557712301026186, 7.30209499879233676382938122844, 8.406693231176218749553958013660, 8.991797058531261839995299601227, 9.353677272744523833793081283445, 9.849000502889582452239528035307, 10.65772235316585565648228861879, 11.37304074217237000786930589773, 11.39858744034346832621337418871, 12.10984816238046023892656484342, 12.75164762652578582436312757752, 13.10547305650341557205541427777, 13.87282216963460635583314164429, 14.80902644681530851113949964493

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