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 + 8·11-s + 16-s − 8·19-s − 4·29-s + 12·41-s − 8·44-s − 49-s + 24·59-s + 28·61-s − 64-s + 16·71-s + 8·76-s − 32·79-s + 20·89-s − 12·101-s − 28·109-s + 4·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/2·4-s + 2.41·11-s + 1/4·16-s − 1.83·19-s − 0.742·29-s + 1.87·41-s − 1.20·44-s − 1/7·49-s + 3.12·59-s + 3.58·61-s − 1/8·64-s + 1.89·71-s + 0.917·76-s − 3.60·79-s + 2.11·89-s − 1.19·101-s − 2.68·109-s + 0.371·116-s + 2.36·121-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 + ⋯

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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.526092498$
$L(\frac12)$  $\approx$  $2.526092498$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.826720435876726295317263037843, −8.647070953951551054909897349010, −8.212943581389159326109155280598, −7.85087508873386847634823185653, −7.33013367924308343034170547799, −6.73308660960340395096537949757, −6.67989243994877390682436296510, −6.45051981026895309423706895960, −5.78842418283782846665269507767, −5.45177830962152926500253079278, −5.13370250830456321850211520648, −4.31857800866127595740348614604, −4.13904590080917166872358906659, −3.79992080817952944918219692036, −3.71753193214688542609658064791, −2.66116301738036995538247288159, −2.35889104555269450690370290147, −1.73257711146949815551442522475, −1.13239738017760799271677146140, −0.56464863805919409306505491886, 0.56464863805919409306505491886, 1.13239738017760799271677146140, 1.73257711146949815551442522475, 2.35889104555269450690370290147, 2.66116301738036995538247288159, 3.71753193214688542609658064791, 3.79992080817952944918219692036, 4.13904590080917166872358906659, 4.31857800866127595740348614604, 5.13370250830456321850211520648, 5.45177830962152926500253079278, 5.78842418283782846665269507767, 6.45051981026895309423706895960, 6.67989243994877390682436296510, 6.73308660960340395096537949757, 7.33013367924308343034170547799, 7.85087508873386847634823185653, 8.212943581389159326109155280598, 8.647070953951551054909897349010, 8.826720435876726295317263037843

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