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  =  $(4,\ 3211264,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.208215220$
$L(\frac12)$  $\approx$  $2.208215220$
$L(\frac{3}{2})$   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

−9.534787196938082010907880759058, −9.098578028097509415707346161483, −8.787882362901984389768593964098, −8.147052255068964491050557395691, −7.969755220518055100671528487094, −7.49402247491036111358051913974, −7.10244484516959338926717268054, −6.66590458914084282251461327277, −6.42260107548125418837606558146, −5.87383896144609479138352216976, −5.22685477394171817252048297576, −5.13872359531302107959415524867, −4.73233704940919964481830423819, −3.73128267301782271183106888680, −3.67724563045336390228385423503, −3.05986235449912193549632571266, −2.88170665607781651375695803383, −1.68724307891371138195539183749, −1.46859891296255733224064010658, −0.58402828276682568742413669358, 0.58402828276682568742413669358, 1.46859891296255733224064010658, 1.68724307891371138195539183749, 2.88170665607781651375695803383, 3.05986235449912193549632571266, 3.67724563045336390228385423503, 3.73128267301782271183106888680, 4.73233704940919964481830423819, 5.13872359531302107959415524867, 5.22685477394171817252048297576, 5.87383896144609479138352216976, 6.42260107548125418837606558146, 6.66590458914084282251461327277, 7.10244484516959338926717268054, 7.49402247491036111358051913974, 7.969755220518055100671528487094, 8.147052255068964491050557395691, 8.787882362901984389768593964098, 9.098578028097509415707346161483, 9.534787196938082010907880759058

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