Properties

Degree 4
Conductor $ 2^{4} \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  + 12·11-s + 8·19-s + 12·29-s + 16·31-s − 24·41-s − 49-s − 20·61-s − 12·71-s + 8·79-s + 24·89-s + 24·101-s − 28·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 3.61·11-s + 1.83·19-s + 2.22·29-s + 2.87·31-s − 3.74·41-s − 1/7·49-s − 2.56·61-s − 1.42·71-s + 0.900·79-s + 2.54·89-s + 2.38·101-s − 2.68·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-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 + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(39690000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6300} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.756986846\)
\(L(\frac12)\)  \(\approx\)  \(5.756986846\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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

−8.248811662233940481852217313048, −7.990084037998074328060509500241, −7.38827832652121158513470294345, −7.09202293056876309228527824723, −6.58274237140609938530377426043, −6.52017477764552950444850539455, −6.27549974579458084361128064107, −6.01516525757656425775262953203, −5.12358101024344324129832713009, −5.03256300300425476626843118864, −4.43040979552528500862834811718, −4.42735300891593009444764014884, −3.69508927659995564259377443052, −3.48434347903952138590673767717, −2.99850710010784147513581177145, −2.81514690297851522087089700088, −1.66048934383948113939919306771, −1.65603105868881597132470199974, −1.03261491511701919146318739714, −0.74438357632839803568015666113, 0.74438357632839803568015666113, 1.03261491511701919146318739714, 1.65603105868881597132470199974, 1.66048934383948113939919306771, 2.81514690297851522087089700088, 2.99850710010784147513581177145, 3.48434347903952138590673767717, 3.69508927659995564259377443052, 4.42735300891593009444764014884, 4.43040979552528500862834811718, 5.03256300300425476626843118864, 5.12358101024344324129832713009, 6.01516525757656425775262953203, 6.27549974579458084361128064107, 6.52017477764552950444850539455, 6.58274237140609938530377426043, 7.09202293056876309228527824723, 7.38827832652121158513470294345, 7.990084037998074328060509500241, 8.248811662233940481852217313048

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