Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{2} \cdot 5^{2} \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  + 3-s − 5-s + 7-s − 6·11-s − 6·13-s − 15-s + 4·17-s + 19-s + 21-s − 4·23-s − 27-s − 16·29-s + 31-s − 6·33-s − 35-s − 7·37-s − 6·39-s − 12·41-s − 2·43-s + 2·47-s − 6·49-s + 4·51-s − 4·53-s + 6·55-s + 57-s − 8·59-s + 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s − 1.80·11-s − 1.66·13-s − 0.258·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 2.97·29-s + 0.179·31-s − 1.04·33-s − 0.169·35-s − 1.15·37-s − 0.960·39-s − 1.87·41-s − 0.304·43-s + 0.291·47-s − 6/7·49-s + 0.560·51-s − 0.549·53-s + 0.809·55-s + 0.132·57-s − 1.04·59-s + 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2822400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1680} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 2822400,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.2242359689\)
\(L(\frac12)\)  \(\approx\)  \(0.2242359689\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.799017111026352812775492126586, −8.916382870521686932449070626475, −8.911610802079641945430882371073, −8.200444409059272060884939297142, −7.77083021382739542142480606847, −7.71044809374244174557197014840, −7.48584219627726846980685576209, −6.89937541278543341700318299896, −6.45253632322690310653945760658, −5.53813185051108553750958509458, −5.50308691489441016538276865499, −5.08384267299640161839052252141, −4.74646859968446338972797107605, −3.94402396756791447706598707449, −3.63127476110818940837555110046, −3.06330121276965694775061512739, −2.65133548171442626748545072215, −1.98534803725846908512535637551, −1.67127185103252551529350466433, −0.16023828338465719120110060703, 0.16023828338465719120110060703, 1.67127185103252551529350466433, 1.98534803725846908512535637551, 2.65133548171442626748545072215, 3.06330121276965694775061512739, 3.63127476110818940837555110046, 3.94402396756791447706598707449, 4.74646859968446338972797107605, 5.08384267299640161839052252141, 5.50308691489441016538276865499, 5.53813185051108553750958509458, 6.45253632322690310653945760658, 6.89937541278543341700318299896, 7.48584219627726846980685576209, 7.71044809374244174557197014840, 7.77083021382739542142480606847, 8.200444409059272060884939297142, 8.911610802079641945430882371073, 8.916382870521686932449070626475, 9.799017111026352812775492126586

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