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 + 12·19-s − 6·29-s − 14·31-s − 2·41-s − 8·44-s − 49-s + 22·59-s + 26·61-s − 64-s + 16·71-s − 12·76-s − 8·79-s + 28·89-s − 12·101-s + 12·109-s + 6·116-s + 26·121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s + 2.41·11-s + 1/4·16-s + 2.75·19-s − 1.11·29-s − 2.51·31-s − 0.312·41-s − 1.20·44-s − 1/7·49-s + 2.86·59-s + 3.32·61-s − 1/8·64-s + 1.89·71-s − 1.37·76-s − 0.900·79-s + 2.96·89-s − 1.19·101-s + 1.14·109-s + 0.557·116-s + 2.36·121-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 + ⋯

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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.523395364$
$L(\frac12)$  $\approx$  $3.523395364$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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

−8.965519997760575876746827726754, −8.474084536017281232602483803504, −8.315633315099515390678892105422, −7.65607057226401001993405753570, −7.20572419112854146307497023922, −7.10726969544952645154602973998, −6.76911569919773997800164190505, −6.25433060652355187640828332068, −5.56798130957128575493504644335, −5.51999883862097348412744905372, −5.21194814994822844272447324282, −4.61089668806072020829715490227, −3.90985887244243575684573922447, −3.75260401564923922740596232305, −3.55955806459958725413863634078, −3.05351979585591314046285291365, −1.98513236476796832385261488948, −1.91802078772026257114788084570, −0.957964923863601312156335678334, −0.77860920818406676750222499911, 0.77860920818406676750222499911, 0.957964923863601312156335678334, 1.91802078772026257114788084570, 1.98513236476796832385261488948, 3.05351979585591314046285291365, 3.55955806459958725413863634078, 3.75260401564923922740596232305, 3.90985887244243575684573922447, 4.61089668806072020829715490227, 5.21194814994822844272447324282, 5.51999883862097348412744905372, 5.56798130957128575493504644335, 6.25433060652355187640828332068, 6.76911569919773997800164190505, 7.10726969544952645154602973998, 7.20572419112854146307497023922, 7.65607057226401001993405753570, 8.315633315099515390678892105422, 8.474084536017281232602483803504, 8.965519997760575876746827726754

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