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 + 16-s − 4·19-s + 6·29-s − 2·31-s + 6·41-s − 49-s + 6·59-s − 2·61-s − 64-s + 4·76-s − 16·79-s + 12·89-s + 36·101-s − 4·109-s − 6·116-s − 22·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 6·164-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/4·16-s − 0.917·19-s + 1.11·29-s − 0.359·31-s + 0.937·41-s − 1/7·49-s + 0.781·59-s − 0.256·61-s − 1/8·64-s + 0.458·76-s − 1.80·79-s + 1.27·89-s + 3.58·101-s − 0.383·109-s − 0.557·116-s − 2·121-s + 0.179·124-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 − 0.468·164-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.777029493$
$L(\frac12)$  $\approx$  $1.777029493$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.892873197570309396618935235142, −8.514491285592630417106541837167, −8.143699875383429433307235246115, −7.86850025021864003922919082565, −7.31353699695357714789958586425, −7.11688400372109128256356569296, −6.43572334442883342116061027384, −6.38263968118976671161944492532, −5.73699273207765516982578236559, −5.56546816637444679235070848430, −4.78443066126930797220887524463, −4.78183526035563087960705749112, −4.09276954009409360383077629486, −3.97704346989484432646435547383, −3.16558000415867748122058097079, −2.99458037291442977361376496155, −2.22236095699846338261164221695, −1.88979011876853662396838087607, −1.06717930086113082484273531165, −0.47440218472797002822762346911, 0.47440218472797002822762346911, 1.06717930086113082484273531165, 1.88979011876853662396838087607, 2.22236095699846338261164221695, 2.99458037291442977361376496155, 3.16558000415867748122058097079, 3.97704346989484432646435547383, 4.09276954009409360383077629486, 4.78183526035563087960705749112, 4.78443066126930797220887524463, 5.56546816637444679235070848430, 5.73699273207765516982578236559, 6.38263968118976671161944492532, 6.43572334442883342116061027384, 7.11688400372109128256356569296, 7.31353699695357714789958586425, 7.86850025021864003922919082565, 8.143699875383429433307235246115, 8.514491285592630417106541837167, 8.892873197570309396618935235142

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