Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{3} \cdot 23^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 7-s − 4·8-s − 2·9-s − 2·14-s + 5·16-s + 12·17-s + 4·18-s + 4·19-s − 10·25-s + 3·28-s − 12·29-s − 6·32-s − 24·34-s − 6·36-s − 8·38-s + 49-s + 20·50-s − 4·56-s + 24·58-s + 16·61-s − 2·63-s + 7·64-s + 36·68-s + 8·72-s + 12·76-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 2/3·9-s − 0.534·14-s + 5/4·16-s + 2.91·17-s + 0.942·18-s + 0.917·19-s − 2·25-s + 0.566·28-s − 2.22·29-s − 1.06·32-s − 4.11·34-s − 36-s − 1.29·38-s + 1/7·49-s + 2.82·50-s − 0.534·56-s + 3.15·58-s + 2.04·61-s − 0.251·63-s + 7/8·64-s + 4.36·68-s + 0.942·72-s + 1.37·76-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 725788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 725788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(725788\)    =    \(2^{2} \cdot 7^{3} \cdot 23^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{725788} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 725788,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−8.110149332059267353023137382824, −7.59769760654238825164816496129, −7.57571100088867902110310233811, −7.01062402394523129137998185488, −6.33381612131422841357801936519, −5.57928681742950427486583645839, −5.57132308384623995366860612457, −5.32616093306835891028088920065, −3.97448209386129731451102084779, −3.73919187382497721540937213023, −3.06849498900201108772759649392, −2.49577833291038984931506482388, −1.63743099327135487468573944042, −1.18173816534949059288556508761, 0, 1.18173816534949059288556508761, 1.63743099327135487468573944042, 2.49577833291038984931506482388, 3.06849498900201108772759649392, 3.73919187382497721540937213023, 3.97448209386129731451102084779, 5.32616093306835891028088920065, 5.57132308384623995366860612457, 5.57928681742950427486583645839, 6.33381612131422841357801936519, 7.01062402394523129137998185488, 7.57571100088867902110310233811, 7.59769760654238825164816496129, 8.110149332059267353023137382824

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