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  − 4-s − 12·11-s + 16-s + 8·19-s + 6·29-s + 10·31-s − 18·41-s + 12·44-s − 49-s + 18·59-s + 22·61-s − 64-s + 24·71-s − 8·76-s + 20·79-s − 12·89-s + 24·101-s − 16·109-s − 6·116-s + 86·121-s − 10·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 3.61·11-s + 1/4·16-s + 1.83·19-s + 1.11·29-s + 1.79·31-s − 2.81·41-s + 1.80·44-s − 1/7·49-s + 2.34·59-s + 2.81·61-s − 1/8·64-s + 2.84·71-s − 0.917·76-s + 2.25·79-s − 1.27·89-s + 2.38·101-s − 1.53·109-s − 0.557·116-s + 7.81·121-s − 0.898·124-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 + ⋯

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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.811667482$
$L(\frac12)$  $\approx$  $1.811667482$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.484416722006998457196752843412, −8.396890690561103216066728692385, −8.149724297132105758874691392591, −8.028024437350136986872283569691, −7.28176097123705725341777601123, −7.23548241246269076465242685135, −6.59662262362520316785158338603, −6.32047192516003649063087768939, −5.44546515561877733406635975891, −5.24929416754664928317637140974, −5.22043629582227355792129521072, −4.92710562915008985118549109562, −4.26477065713028948207898379877, −3.66196077915573327863344971750, −3.05783996250439134404285734861, −3.01192924951030813575281827571, −2.30977461324972319335074126966, −2.05084966473532951062737151288, −0.801377077126200488323559083344, −0.58985917366547297451182552514, 0.58985917366547297451182552514, 0.801377077126200488323559083344, 2.05084966473532951062737151288, 2.30977461324972319335074126966, 3.01192924951030813575281827571, 3.05783996250439134404285734861, 3.66196077915573327863344971750, 4.26477065713028948207898379877, 4.92710562915008985118549109562, 5.22043629582227355792129521072, 5.24929416754664928317637140974, 5.44546515561877733406635975891, 6.32047192516003649063087768939, 6.59662262362520316785158338603, 7.23548241246269076465242685135, 7.28176097123705725341777601123, 8.028024437350136986872283569691, 8.149724297132105758874691392591, 8.396890690561103216066728692385, 8.484416722006998457196752843412

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