Properties

Degree 4
Conductor 3319
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 4·3-s + 4·4-s − 5·5-s + 12·6-s − 3·7-s − 3·8-s + 8·9-s + 15·10-s − 2·11-s − 16·12-s − 4·13-s + 9·14-s + 20·15-s + 3·16-s − 2·17-s − 24·18-s − 20·20-s + 12·21-s + 6·22-s − 23-s + 12·24-s + 12·25-s + 12·26-s − 12·27-s − 12·28-s − 8·29-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 2·4-s − 2.23·5-s + 4.89·6-s − 1.13·7-s − 1.06·8-s + 8/3·9-s + 4.74·10-s − 0.603·11-s − 4.61·12-s − 1.10·13-s + 2.40·14-s + 5.16·15-s + 3/4·16-s − 0.485·17-s − 5.65·18-s − 4.47·20-s + 2.61·21-s + 1.27·22-s − 0.208·23-s + 2.44·24-s + 12/5·25-s + 2.35·26-s − 2.30·27-s − 2.26·28-s − 1.48·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3319 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3319 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3319\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3319} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 3319,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3319$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 3319$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3319$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 74 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 28 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T - p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 11 T + 65 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 154 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 5 T + 26 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 32 T + 447 T^{2} + 32 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.3938479658, −18.2580273292, −17.4645704872, −16.9879156301, −16.7814047905, −16.4058573769, −15.7955123936, −15.3310198499, −15.1226319001, −13.6247606441, −12.6576755659, −12.2114837333, −12.0193009042, −11.2005691286, −10.9981825876, −10.3483485727, −9.76512858602, −9.18509401409, −8.24396548524, −7.71994464966, −7.16377749696, −6.56064873395, −5.57423781475, −4.71287415515, −3.56807591826, 0, 0, 3.56807591826, 4.71287415515, 5.57423781475, 6.56064873395, 7.16377749696, 7.71994464966, 8.24396548524, 9.18509401409, 9.76512858602, 10.3483485727, 10.9981825876, 11.2005691286, 12.0193009042, 12.2114837333, 12.6576755659, 13.6247606441, 15.1226319001, 15.3310198499, 15.7955123936, 16.4058573769, 16.7814047905, 16.9879156301, 17.4645704872, 18.2580273292, 18.3938479658

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