Properties

Degree 4
Conductor $ 2^{16} \cdot 7^{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  + 2·7-s + 2·9-s + 12·17-s + 10·25-s + 8·31-s − 12·41-s + 24·47-s + 3·49-s + 4·63-s − 4·73-s − 16·79-s − 5·81-s + 12·89-s − 20·97-s − 8·103-s + 12·113-s + 24·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 0.755·7-s + 2/3·9-s + 2.91·17-s + 2·25-s + 1.43·31-s − 1.87·41-s + 3.50·47-s + 3/7·49-s + 0.503·63-s − 0.468·73-s − 1.80·79-s − 5/9·81-s + 1.27·89-s − 2.03·97-s − 0.788·103-s + 1.12·113-s + 2.20·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3211264\)    =    \(2^{16} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1792} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3211264,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.953085809$
$L(\frac12)$  $\approx$  $3.953085809$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
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

−9.582244117207813306484081522867, −8.969894990703285873460521721506, −8.629298718029248324586048066036, −8.319348796562183905995251752788, −7.942853529295811248914099066303, −7.42972218565900774766647327548, −7.10860822769963937535741884241, −6.98830406054245652288390218989, −6.12207400655819751030723132536, −5.89274378722533440007963278070, −5.22322005488711877978514198455, −5.20561269830225679526579033922, −4.47867284023329427029777746450, −4.20828599827424517765801743325, −3.53830435246007457620500860630, −3.05346149573610183813753309475, −2.70743521323646971641217695588, −1.85911327474741436580407073794, −1.07240091572505906501474742470, −1.00373239285767889559039585838, 1.00373239285767889559039585838, 1.07240091572505906501474742470, 1.85911327474741436580407073794, 2.70743521323646971641217695588, 3.05346149573610183813753309475, 3.53830435246007457620500860630, 4.20828599827424517765801743325, 4.47867284023329427029777746450, 5.20561269830225679526579033922, 5.22322005488711877978514198455, 5.89274378722533440007963278070, 6.12207400655819751030723132536, 6.98830406054245652288390218989, 7.10860822769963937535741884241, 7.42972218565900774766647327548, 7.942853529295811248914099066303, 8.319348796562183905995251752788, 8.629298718029248324586048066036, 8.969894990703285873460521721506, 9.582244117207813306484081522867

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