Properties

Degree $4$
Conductor $3887$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s − 2·9-s − 2·12-s − 2·13-s − 3·16-s − 2·17-s + 3·23-s + 6·25-s − 10·27-s + 8·29-s + 2·36-s − 4·39-s + 10·43-s − 6·48-s − 8·49-s − 4·51-s + 2·52-s − 8·53-s + 7·64-s + 2·68-s + 6·69-s + 12·75-s − 14·79-s − 5·81-s + 16·87-s − 3·92-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 2/3·9-s − 0.577·12-s − 0.554·13-s − 3/4·16-s − 0.485·17-s + 0.625·23-s + 6/5·25-s − 1.92·27-s + 1.48·29-s + 1/3·36-s − 0.640·39-s + 1.52·43-s − 0.866·48-s − 8/7·49-s − 0.560·51-s + 0.277·52-s − 1.09·53-s + 7/8·64-s + 0.242·68-s + 0.722·69-s + 1.38·75-s − 1.57·79-s − 5/9·81-s + 1.71·87-s − 0.312·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3887\)    =    \(13^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3887} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3887,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9009011983\)
\(L(\frac12)\) \(\approx\) \(0.9009011983\)
\(L(\frac{3}{2})\) 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)$
bad13$C_2$ \( 1 + 2 T + p T^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 170 T^{2} + p^{2} T^{4} \)
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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

−12.73533774163262736763912873714, −11.99194214963161824650437188889, −11.26475990806612486488139835344, −10.84758190365504273016954464430, −9.929652181218310602263282178320, −9.311341964711249074616543068169, −8.781296642082660196397410944930, −8.496631523612372874930816564504, −7.72629502163803240846402078344, −6.92983547406076020553289617152, −6.13771170368324671342892193474, −5.06169023374393873111537198259, −4.42013484455619515379930105876, −3.17090161637365476303900486433, −2.51569256430844859340267012894, 2.51569256430844859340267012894, 3.17090161637365476303900486433, 4.42013484455619515379930105876, 5.06169023374393873111537198259, 6.13771170368324671342892193474, 6.92983547406076020553289617152, 7.72629502163803240846402078344, 8.496631523612372874930816564504, 8.781296642082660196397410944930, 9.311341964711249074616543068169, 9.929652181218310602263282178320, 10.84758190365504273016954464430, 11.26475990806612486488139835344, 11.99194214963161824650437188889, 12.73533774163262736763912873714

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