Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} \cdot 61^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4-s + 6·9-s − 4·12-s − 8·13-s + 16-s + 4·19-s − 10·25-s + 4·27-s + 6·36-s + 32·39-s + 12·41-s − 24·47-s − 4·48-s + 49-s − 8·52-s − 16·57-s + 8·61-s + 64-s + 4·73-s + 40·75-s + 4·76-s − 37·81-s − 12·83-s − 20·97-s − 10·100-s − 8·103-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 2·9-s − 1.15·12-s − 2.21·13-s + 1/4·16-s + 0.917·19-s − 2·25-s + 0.769·27-s + 36-s + 5.12·39-s + 1.87·41-s − 3.50·47-s − 0.577·48-s + 1/7·49-s − 1.10·52-s − 2.11·57-s + 1.02·61-s + 1/8·64-s + 0.468·73-s + 4.61·75-s + 0.458·76-s − 4.11·81-s − 1.31·83-s − 2.03·97-s − 100-s − 0.788·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(729316\)    =    \(2^{2} \cdot 7^{2} \cdot 61^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{729316} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 729316,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2513182106$
$L(\frac12)$  $\approx$  $0.2513182106$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−8.173449077566698438751222908068, −7.57571100088867902110310233811, −7.33566132154895538455272247406, −6.86227272075079610412028702533, −6.34728331211217856949764881281, −6.01523816258371890896044883784, −5.57928681742950427486583645839, −5.17900673049925304081269440708, −4.85492146999244580615110535351, −4.38080265430113012265676461865, −3.56574648496591683143026739238, −2.79745735406815554125194503256, −2.27907197519064804199458946410, −1.34430444523648923531944095258, −0.28992103454452119940322760112, 0.28992103454452119940322760112, 1.34430444523648923531944095258, 2.27907197519064804199458946410, 2.79745735406815554125194503256, 3.56574648496591683143026739238, 4.38080265430113012265676461865, 4.85492146999244580615110535351, 5.17900673049925304081269440708, 5.57928681742950427486583645839, 6.01523816258371890896044883784, 6.34728331211217856949764881281, 6.86227272075079610412028702533, 7.33566132154895538455272247406, 7.57571100088867902110310233811, 8.173449077566698438751222908068

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