Properties

Degree 4
Conductor $ 2^{6} \cdot 3 \cdot 17 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s + 6·6-s + 4·9-s − 6·11-s − 6·12-s − 4·16-s − 7·17-s − 8·18-s − 4·19-s + 12·22-s − 2·25-s + 8·32-s + 18·33-s + 14·34-s + 8·36-s + 8·38-s − 4·41-s + 8·43-s − 12·44-s + 12·48-s − 2·49-s + 4·50-s + 21·51-s + 12·57-s − 4·59-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s + 4/3·9-s − 1.80·11-s − 1.73·12-s − 16-s − 1.69·17-s − 1.88·18-s − 0.917·19-s + 2.55·22-s − 2/5·25-s + 1.41·32-s + 3.13·33-s + 2.40·34-s + 4/3·36-s + 1.29·38-s − 0.624·41-s + 1.21·43-s − 1.80·44-s + 1.73·48-s − 2/7·49-s + 0.565·50-s + 2.94·51-s + 1.58·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3264\)    =    \(2^{6} \cdot 3 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{3264} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 3264,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 6 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−12.21042573364748676074371926432, −11.51872936506842932746880271040, −10.98133269166747767917912100925, −10.61403928556650556894125675254, −10.33963011208683484979467530412, −9.438558187505883569273911834359, −8.733721189989508905862760810801, −8.082402023422120616644237332671, −7.38360144889446860906003907987, −6.62198346346856840746921529147, −6.01678572040208163918026990344, −5.08860375066156614816012066413, −4.46631018668298863468822429117, −2.33446241377498533200596611163, 0, 2.33446241377498533200596611163, 4.46631018668298863468822429117, 5.08860375066156614816012066413, 6.01678572040208163918026990344, 6.62198346346856840746921529147, 7.38360144889446860906003907987, 8.082402023422120616644237332671, 8.733721189989508905862760810801, 9.438558187505883569273911834359, 10.33963011208683484979467530412, 10.61403928556650556894125675254, 10.98133269166747767917912100925, 11.51872936506842932746880271040, 12.21042573364748676074371926432

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