Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \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 + 8·19-s − 2·25-s + 4·27-s − 4·29-s + 12·37-s − 16·47-s − 12·53-s + 16·57-s + 24·59-s − 4·75-s + 5·81-s − 8·83-s − 8·87-s − 32·103-s − 20·109-s + 24·111-s + 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 32·141-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 1.83·19-s − 2/5·25-s + 0.769·27-s − 0.742·29-s + 1.97·37-s − 2.33·47-s − 1.64·53-s + 2.11·57-s + 3.12·59-s − 0.461·75-s + 5/9·81-s − 0.878·83-s − 0.857·87-s − 3.15·103-s − 1.91·109-s + 2.27·111-s + 3.38·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.69·141-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9408} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.185852051\)
\(L(\frac12)\)  \(\approx\)  \(5.185852051\)
\(L(\frac{3}{2})\)   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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 170 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−7.72422264843946343694557146770, −7.71805484278640167853130520231, −7.34415402566031024099920148754, −6.91027350984720031143776283548, −6.45938602286487460470251211713, −6.44047781134068395470670019815, −5.61256073191123824520690500173, −5.56338792392088227219697612548, −5.11148135797121067663999258614, −4.75829742207552320811575957197, −4.14866656069026237825313021656, −4.09375816188670875867908189525, −3.51092871880825316587160258717, −3.24941329787843239880586863394, −2.70643966111150143119198738135, −2.68967747548855150321070991231, −1.86410960555767849634493943530, −1.64808447400130624722306976293, −1.07740963425290224405502400682, −0.49401376832659698610873216506, 0.49401376832659698610873216506, 1.07740963425290224405502400682, 1.64808447400130624722306976293, 1.86410960555767849634493943530, 2.68967747548855150321070991231, 2.70643966111150143119198738135, 3.24941329787843239880586863394, 3.51092871880825316587160258717, 4.09375816188670875867908189525, 4.14866656069026237825313021656, 4.75829742207552320811575957197, 5.11148135797121067663999258614, 5.56338792392088227219697612548, 5.61256073191123824520690500173, 6.44047781134068395470670019815, 6.45938602286487460470251211713, 6.91027350984720031143776283548, 7.34415402566031024099920148754, 7.71805484278640167853130520231, 7.72422264843946343694557146770

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