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 − 8·25-s − 4·27-s + 8·31-s − 8·37-s + 24·47-s − 20·53-s + 16·75-s + 5·81-s − 8·83-s − 16·93-s + 24·103-s + 8·109-s + 16·111-s − 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 48·141-s + 149-s + 151-s + 157-s + 40·159-s + 163-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 8/5·25-s − 0.769·27-s + 1.43·31-s − 1.31·37-s + 3.50·47-s − 2.74·53-s + 1.84·75-s + 5/9·81-s − 0.878·83-s − 1.65·93-s + 2.36·103-s + 0.766·109-s + 1.51·111-s − 0.376·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.04·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 3.17·159-s + 0.0783·163-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  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.399656483\)
\(L(\frac12)\)  \(\approx\)  \(1.399656483\)
\(L(\frac{3}{2})\)   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$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
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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

−7.65073084407784411736313931587, −7.51459575424776855093613593925, −7.23429679082443462535994273216, −6.79233857099951129486591792206, −6.37135339811430415999230939353, −6.20250335783729617003269280733, −5.74342477355179153731959925307, −5.66781468334466374615626317060, −5.10647592199974558494119292523, −4.82554840487620804777992537324, −4.39048783354774690280236127702, −4.20246274677335008777852687988, −3.56701637997582601146434517321, −3.49106880330297909021481093433, −2.63379032368089645045389597907, −2.51901954319678542874303631599, −1.68751951739619551327083149121, −1.59364298498603317639862260818, −0.78096710584674172255292214303, −0.38356289442500274545899260156, 0.38356289442500274545899260156, 0.78096710584674172255292214303, 1.59364298498603317639862260818, 1.68751951739619551327083149121, 2.51901954319678542874303631599, 2.63379032368089645045389597907, 3.49106880330297909021481093433, 3.56701637997582601146434517321, 4.20246274677335008777852687988, 4.39048783354774690280236127702, 4.82554840487620804777992537324, 5.10647592199974558494119292523, 5.66781468334466374615626317060, 5.74342477355179153731959925307, 6.20250335783729617003269280733, 6.37135339811430415999230939353, 6.79233857099951129486591792206, 7.23429679082443462535994273216, 7.51459575424776855093613593925, 7.65073084407784411736313931587

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