Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{6} \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 + 2·17-s + 10·23-s + 6·25-s + 14·31-s + 4·41-s + 3·49-s + 30·71-s + 28·79-s − 26·89-s − 28·97-s + 6·103-s + 12·113-s + 4·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 20·161-s + 163-s + 167-s + 169-s + 173-s + ⋯
L(s)  = 1  + 0.755·7-s + 0.485·17-s + 2.08·23-s + 6/5·25-s + 2.51·31-s + 0.624·41-s + 3/7·49-s + 3.56·71-s + 3.15·79-s − 2.75·89-s − 2.84·97-s + 0.591·103-s + 1.12·113-s + 0.366·119-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 + 0.0798·157-s + 1.57·161-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36578304\)    =    \(2^{10} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 36578304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $5.032795417$
$L(\frac12)$  $\approx$  $5.032795417$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.160121445874212267566844211224, −8.129666030689296445274998829153, −7.43015231496483898726799062342, −7.29296081509303595524054842237, −6.75885488534285347256875211226, −6.55062231685487687322961879923, −6.27963530035108298059024491788, −5.64927597714867348917153816715, −5.21006649429812336758834395192, −5.12834744773377447375616099912, −4.53267727581114999926143974025, −4.52744584124502477203492833870, −3.76819757261829282929161123519, −3.46982435331385974920950612120, −2.77789726791239328209640723026, −2.75163516912380957984230910762, −2.16884720685062768093969796126, −1.46644659443831966827454615355, −0.890840586339115948092554026183, −0.78187672482828431219914192437, 0.78187672482828431219914192437, 0.890840586339115948092554026183, 1.46644659443831966827454615355, 2.16884720685062768093969796126, 2.75163516912380957984230910762, 2.77789726791239328209640723026, 3.46982435331385974920950612120, 3.76819757261829282929161123519, 4.52744584124502477203492833870, 4.53267727581114999926143974025, 5.12834744773377447375616099912, 5.21006649429812336758834395192, 5.64927597714867348917153816715, 6.27963530035108298059024491788, 6.55062231685487687322961879923, 6.75885488534285347256875211226, 7.29296081509303595524054842237, 7.43015231496483898726799062342, 8.129666030689296445274998829153, 8.160121445874212267566844211224

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