Properties

Degree 4
Conductor $ 2^{8} \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  + 4·7-s − 4·19-s + 7·25-s − 12·29-s + 4·31-s + 2·37-s + 6·47-s + 9·49-s − 24·53-s − 6·59-s + 18·83-s + 16·103-s − 14·109-s − 12·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.917·19-s + 7/5·25-s − 2.22·29-s + 0.718·31-s + 0.328·37-s + 0.875·47-s + 9/7·49-s − 3.29·53-s − 0.781·59-s + 1.97·83-s + 1.57·103-s − 1.34·109-s − 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.533109085\)
\(L(\frac12)\)  \(\approx\)  \(2.533109085\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 155 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 182 T^{2} + p^{2} T^{4} \)
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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

−9.174971042401283626012991911753, −8.454369367263264743566368750137, −8.133164509530775991616523688598, −7.79341856962824814822078639654, −7.57435919717308253093760719217, −7.08510638329960507782034790136, −6.67284401114675792687970826959, −6.17456228098049758001621172680, −5.96806521397015346815143852771, −5.34334659418224474882531255050, −5.02426904633694317362668116793, −4.57845852624762671886782080718, −4.46108971332335328465375995974, −3.74763002946318462915919133551, −3.41495188650871206208726523391, −2.74194792590763441872372261865, −2.28877149500466420519370970972, −1.69258874758613181219641855651, −1.40009139963704119765595218879, −0.50244613608157697575908991457, 0.50244613608157697575908991457, 1.40009139963704119765595218879, 1.69258874758613181219641855651, 2.28877149500466420519370970972, 2.74194792590763441872372261865, 3.41495188650871206208726523391, 3.74763002946318462915919133551, 4.46108971332335328465375995974, 4.57845852624762671886782080718, 5.02426904633694317362668116793, 5.34334659418224474882531255050, 5.96806521397015346815143852771, 6.17456228098049758001621172680, 6.67284401114675792687970826959, 7.08510638329960507782034790136, 7.57435919717308253093760719217, 7.79341856962824814822078639654, 8.133164509530775991616523688598, 8.454369367263264743566368750137, 9.174971042401283626012991911753

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