Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{4} \cdot 5^{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  + 2·2-s + 3·4-s − 2·7-s + 4·8-s + 4·13-s − 4·14-s + 5·16-s − 4·17-s + 8·19-s + 4·23-s + 8·26-s − 6·28-s − 4·29-s + 8·31-s + 6·32-s − 8·34-s + 4·37-s + 16·38-s + 12·41-s + 8·43-s + 8·46-s − 8·47-s + 3·49-s + 12·52-s + 12·53-s − 8·56-s − 8·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s + 1.10·13-s − 1.06·14-s + 5/4·16-s − 0.970·17-s + 1.83·19-s + 0.834·23-s + 1.56·26-s − 1.13·28-s − 0.742·29-s + 1.43·31-s + 1.06·32-s − 1.37·34-s + 0.657·37-s + 2.59·38-s + 1.87·41-s + 1.21·43-s + 1.17·46-s − 1.16·47-s + 3/7·49-s + 1.66·52-s + 1.64·53-s − 1.06·56-s − 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9922500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $8.818038026$
$L(\frac12)$  $\approx$  $8.818038026$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 128 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−8.718500198785599827768234947999, −8.671501638574013276719841265501, −7.888616375787963028727798733286, −7.60042568897851394711821255594, −7.30888466539120662197892038024, −6.90059599289980208404241258947, −6.32809171192839689921668176380, −6.30278614976401395409516296359, −5.81661532101620633712824348602, −5.45890174961157953891347456045, −4.97905503640449219249066760321, −4.68742545986080523005139201866, −4.00625611932096696610024124243, −3.88434872757924766026719053158, −3.38127715336835055545745174316, −2.94152371679574146245178227398, −2.49277899131395804471994601550, −2.12373740570617103515999382046, −1.05784179025911638201657044668, −0.886049184299352588669362709967, 0.886049184299352588669362709967, 1.05784179025911638201657044668, 2.12373740570617103515999382046, 2.49277899131395804471994601550, 2.94152371679574146245178227398, 3.38127715336835055545745174316, 3.88434872757924766026719053158, 4.00625611932096696610024124243, 4.68742545986080523005139201866, 4.97905503640449219249066760321, 5.45890174961157953891347456045, 5.81661532101620633712824348602, 6.30278614976401395409516296359, 6.32809171192839689921668176380, 6.90059599289980208404241258947, 7.30888466539120662197892038024, 7.60042568897851394711821255594, 7.888616375787963028727798733286, 8.671501638574013276719841265501, 8.718500198785599827768234947999

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