Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} \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  + 5·7-s + 19-s − 5·25-s + 11·31-s + 11·37-s + 18·49-s − 12·61-s + 27·67-s + 3·73-s + 9·79-s − 13·103-s + 19·109-s − 11·121-s + 127-s + 131-s + 5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 25·175-s + 179-s + ⋯
L(s)  = 1  + 1.88·7-s + 0.229·19-s − 25-s + 1.97·31-s + 1.80·37-s + 18/7·49-s − 1.53·61-s + 3.29·67-s + 0.351·73-s + 1.01·79-s − 1.28·103-s + 1.81·109-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.433·133-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 − 0.0769·169-s + 0.0760·173-s − 1.88·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.936669388\)
\(L(\frac12)\)  \(\approx\)  \(2.936669388\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.04369075740283898964419154714, −9.907146378929162580932039969213, −9.307985346581686852034792200507, −8.908570292508444541811797263013, −8.353888459261358224880990529514, −8.084055615272246010021346641484, −7.69646155047672918152114601450, −7.56052952340716718020469402504, −6.67589517465499948367453876473, −6.44295782931424713837663354124, −5.79249863017645001869900063218, −5.42368296416434588925194671934, −4.74671477932337098703228579638, −4.67850642741028613609107877559, −4.01589421443877902256444259515, −3.54829624376638419107881016540, −2.53215073632567230730026855483, −2.34767897603019312967976076449, −1.45580002141998516231787664156, −0.872390336948986254722597598955, 0.872390336948986254722597598955, 1.45580002141998516231787664156, 2.34767897603019312967976076449, 2.53215073632567230730026855483, 3.54829624376638419107881016540, 4.01589421443877902256444259515, 4.67850642741028613609107877559, 4.74671477932337098703228579638, 5.42368296416434588925194671934, 5.79249863017645001869900063218, 6.44295782931424713837663354124, 6.67589517465499948367453876473, 7.56052952340716718020469402504, 7.69646155047672918152114601450, 8.084055615272246010021346641484, 8.353888459261358224880990529514, 8.908570292508444541811797263013, 9.307985346581686852034792200507, 9.907146378929162580932039969213, 10.04369075740283898964419154714

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