Properties

Degree 4
Conductor $ 2^{8} \cdot 7^{4} $
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  + 2·3-s + 3·9-s + 8·13-s + 6·17-s − 2·19-s + 5·25-s + 10·27-s − 12·29-s + 4·31-s − 2·37-s + 16·39-s − 12·41-s − 16·43-s + 12·47-s + 12·51-s − 6·53-s − 4·57-s + 6·59-s + 8·61-s − 4·67-s + 2·73-s + 10·75-s + 8·79-s + 20·81-s − 12·83-s − 24·87-s − 6·89-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 2.21·13-s + 1.45·17-s − 0.458·19-s + 25-s + 1.92·27-s − 2.22·29-s + 0.718·31-s − 0.328·37-s + 2.56·39-s − 1.87·41-s − 2.43·43-s + 1.75·47-s + 1.68·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s + 1.02·61-s − 0.488·67-s + 0.234·73-s + 1.15·75-s + 0.900·79-s + 20/9·81-s − 1.31·83-s − 2.57·87-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(614656\)    =    \(2^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{784} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 614656,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.78551$
$L(\frac12)$  $\approx$  $3.78551$
$L(\frac{3}{2})$   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 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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

−10.34586967778408609110973286532, −10.06274249136189414685461143675, −9.707851007875754713392808853370, −9.064425253230624987589428692824, −8.666227011445378470037205584876, −8.332237409188002119946297277530, −8.329005074772606692744430179159, −7.54537155574140241934656446245, −7.00986246105369453839216729199, −6.81573567429567800497384834958, −5.97583003985655602163386035513, −5.86978903777549079274329894396, −4.99345678444641967411552541619, −4.68293780932496330597192328397, −3.74433925007353034501583347528, −3.40678987924838441751322896699, −3.35798242908381928438757915636, −2.33030116372514581166531426270, −1.62450956709714827554626904609, −1.04073954424139627738905894688, 1.04073954424139627738905894688, 1.62450956709714827554626904609, 2.33030116372514581166531426270, 3.35798242908381928438757915636, 3.40678987924838441751322896699, 3.74433925007353034501583347528, 4.68293780932496330597192328397, 4.99345678444641967411552541619, 5.86978903777549079274329894396, 5.97583003985655602163386035513, 6.81573567429567800497384834958, 7.00986246105369453839216729199, 7.54537155574140241934656446245, 8.329005074772606692744430179159, 8.332237409188002119946297277530, 8.666227011445378470037205584876, 9.064425253230624987589428692824, 9.707851007875754713392808853370, 10.06274249136189414685461143675, 10.34586967778408609110973286532

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