Properties

Degree 4
Conductor $ 2^{2} \cdot 7 \cdot 29^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·5-s − 5·7-s − 9-s − 2·13-s + 16-s − 2·20-s − 4·23-s − 7·25-s − 5·28-s − 2·29-s + 10·35-s − 36-s + 2·45-s + 14·49-s − 2·52-s − 6·53-s + 8·59-s + 5·63-s + 64-s + 4·65-s + 4·67-s + 12·71-s − 2·80-s − 8·81-s − 24·83-s + 10·91-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s − 1.88·7-s − 1/3·9-s − 0.554·13-s + 1/4·16-s − 0.447·20-s − 0.834·23-s − 7/5·25-s − 0.944·28-s − 0.371·29-s + 1.69·35-s − 1/6·36-s + 0.298·45-s + 2·49-s − 0.277·52-s − 0.824·53-s + 1.04·59-s + 0.629·63-s + 1/8·64-s + 0.496·65-s + 0.488·67-s + 1.42·71-s − 0.223·80-s − 8/9·81-s − 2.63·83-s + 1.04·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 23548 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 23548 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(23548\)    =    \(2^{2} \cdot 7 \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{23548} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 23548,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 113 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.29549113238609177901960808588, −9.885566847319717954512995670480, −9.610320735229573674162428613672, −8.891520983856759865543945989227, −8.165524035949405987745903348827, −7.72638365652665252679117085675, −7.06795694550111966917162928355, −6.64812387469283597618551809964, −5.93957381300218141915797314580, −5.53376773666531770794204469723, −4.33408715449728664374263643132, −3.71654578176924070556551344061, −3.13708522456824816788458902897, −2.24060991266744443919987085129, 0, 2.24060991266744443919987085129, 3.13708522456824816788458902897, 3.71654578176924070556551344061, 4.33408715449728664374263643132, 5.53376773666531770794204469723, 5.93957381300218141915797314580, 6.64812387469283597618551809964, 7.06795694550111966917162928355, 7.72638365652665252679117085675, 8.165524035949405987745903348827, 8.891520983856759865543945989227, 9.610320735229573674162428613672, 9.885566847319717954512995670480, 10.29549113238609177901960808588

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