Properties

Degree 4
Conductor $ 2^{16} \cdot 3^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 9-s − 12·23-s + 10·25-s − 16·31-s − 24·41-s − 24·47-s + 3·49-s − 2·63-s + 12·71-s + 20·73-s + 8·79-s + 81-s − 24·89-s − 20·97-s + 16·103-s − 12·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 24·161-s + 163-s + ⋯
L(s)  = 1  + 0.755·7-s − 1/3·9-s − 2.50·23-s + 2·25-s − 2.87·31-s − 3.74·41-s − 3.50·47-s + 3/7·49-s − 0.251·63-s + 1.42·71-s + 2.34·73-s + 0.900·79-s + 1/9·81-s − 2.54·89-s − 2.03·97-s + 1.57·103-s − 1.12·113-s − 1.27·121-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 − 1.89·161-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(28901376\)    =    \(2^{16} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{5376} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 28901376,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 + T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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

−7.983081919844388091019770191939, −7.942696774340037832322275123907, −7.16871189149650957205261704801, −6.89595211369618080201968592524, −6.60145928989985901456930619813, −6.39204791689497160412699890303, −5.57844285564960434267849504143, −5.54229606706781697400865913072, −5.02742272292131235884624338243, −4.87548881225291001669457992762, −4.37511585051108608285918087555, −3.72163334974701637072978281481, −3.46445598051237037944234230449, −3.32011788639021068020984106117, −2.44472394532890852593298668173, −2.06875207797482378266165752435, −1.64127100441604991309627763858, −1.29227534929850042669879402250, 0, 0, 1.29227534929850042669879402250, 1.64127100441604991309627763858, 2.06875207797482378266165752435, 2.44472394532890852593298668173, 3.32011788639021068020984106117, 3.46445598051237037944234230449, 3.72163334974701637072978281481, 4.37511585051108608285918087555, 4.87548881225291001669457992762, 5.02742272292131235884624338243, 5.54229606706781697400865913072, 5.57844285564960434267849504143, 6.39204791689497160412699890303, 6.60145928989985901456930619813, 6.89595211369618080201968592524, 7.16871189149650957205261704801, 7.942696774340037832322275123907, 7.983081919844388091019770191939

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