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 − 4·11-s + 16-s − 10·29-s + 14·31-s − 14·41-s + 4·44-s − 49-s − 10·59-s − 6·61-s − 64-s − 24·71-s − 20·79-s − 20·89-s − 24·101-s + 10·116-s − 10·121-s − 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 14·164-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.20·11-s + 1/4·16-s − 1.85·29-s + 2.51·31-s − 2.18·41-s + 0.603·44-s − 1/7·49-s − 1.30·59-s − 0.768·61-s − 1/8·64-s − 2.84·71-s − 2.25·79-s − 2.11·89-s − 2.38·101-s + 0.928·116-s − 0.909·121-s − 1.25·124-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 + 0.0798·157-s + 0.0783·163-s + 1.09·164-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.05742922833$
$L(\frac12)$  $\approx$  $0.05742922833$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
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

−8.883989964723449463694998060460, −8.361993452248401674332663346890, −8.271283347321822959936361704402, −7.66073612755731953698203906627, −7.59487215809284527637507476696, −6.94207816007830559333098286809, −6.75169138636964246514282919024, −6.07461372182509048143826465055, −5.84186049145859006025153687105, −5.43108908830618350397645807681, −5.03052056665136633938027836221, −4.56536236662708124509968998309, −4.35545661260127612915762174986, −3.76247462262771287891751462099, −3.24293870197874569928238469942, −2.76336268455975342400461868637, −2.55795096240255478026131980757, −1.52567084440777478986302697144, −1.40496130894693070090633102398, −0.07440349151446252311737012710, 0.07440349151446252311737012710, 1.40496130894693070090633102398, 1.52567084440777478986302697144, 2.55795096240255478026131980757, 2.76336268455975342400461868637, 3.24293870197874569928238469942, 3.76247462262771287891751462099, 4.35545661260127612915762174986, 4.56536236662708124509968998309, 5.03052056665136633938027836221, 5.43108908830618350397645807681, 5.84186049145859006025153687105, 6.07461372182509048143826465055, 6.75169138636964246514282919024, 6.94207816007830559333098286809, 7.59487215809284527637507476696, 7.66073612755731953698203906627, 8.271283347321822959936361704402, 8.361993452248401674332663346890, 8.883989964723449463694998060460

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