Properties

Degree $4$
Conductor $19663$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 2·7-s − 4·9-s − 4·11-s − 7·13-s + 17-s − 2·25-s − 4·28-s − 3·29-s + 8·36-s + 6·37-s + 4·43-s + 8·44-s + 5·47-s − 4·49-s + 14·52-s − 9·53-s + 59-s − 8·63-s + 8·64-s − 2·68-s − 8·77-s + 7·81-s + 19·89-s − 14·91-s − 21·97-s + 16·99-s + ⋯
L(s)  = 1  − 4-s + 0.755·7-s − 4/3·9-s − 1.20·11-s − 1.94·13-s + 0.242·17-s − 2/5·25-s − 0.755·28-s − 0.557·29-s + 4/3·36-s + 0.986·37-s + 0.609·43-s + 1.20·44-s + 0.729·47-s − 4/7·49-s + 1.94·52-s − 1.23·53-s + 0.130·59-s − 1.00·63-s + 64-s − 0.242·68-s − 0.911·77-s + 7/9·81-s + 2.01·89-s − 1.46·91-s − 2.13·97-s + 1.60·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19663 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19663 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19663\)    =    \(7 \cdot 53^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{19663} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 19663,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - T + p T^{2} ) \)
53$C_2$ \( 1 + 9 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.76305445001311360033750927946, −9.914025875056690197244418347116, −9.524735517365554249650084824259, −9.072025373519788075499494740332, −8.364614745539811994938532187031, −7.81028863098103963352928210080, −7.66503807984763803384391583773, −6.68362429781283151241745289927, −5.70693966374383878035099037322, −5.26755994374952138216311271196, −4.83114909218890702163382477873, −4.13589516725346301733789107742, −2.92839172116471089209329754853, −2.30438251415027751560108839668, 0, 2.30438251415027751560108839668, 2.92839172116471089209329754853, 4.13589516725346301733789107742, 4.83114909218890702163382477873, 5.26755994374952138216311271196, 5.70693966374383878035099037322, 6.68362429781283151241745289927, 7.66503807984763803384391583773, 7.81028863098103963352928210080, 8.364614745539811994938532187031, 9.072025373519788075499494740332, 9.524735517365554249650084824259, 9.914025875056690197244418347116, 10.76305445001311360033750927946

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