Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 8·11-s + 16-s − 20·29-s − 16·31-s + 36-s − 4·41-s + 8·44-s − 49-s − 8·59-s − 12·61-s − 64-s − 24·71-s + 16·79-s + 81-s − 28·89-s + 8·99-s − 12·101-s + 36·109-s + 20·116-s + 26·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s − 3.71·29-s − 2.87·31-s + 1/6·36-s − 0.624·41-s + 1.20·44-s − 1/7·49-s − 1.04·59-s − 1.53·61-s − 1/8·64-s − 2.84·71-s + 1.80·79-s + 1/9·81-s − 2.96·89-s + 0.804·99-s − 1.19·101-s + 3.44·109-s + 1.85·116-s + 2.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1102500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1050} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 1102500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T^{2} \)
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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−9.525039483962604332555282034981, −9.323845620263769474202194793591, −8.998605946800609538118590352237, −8.476667599480528572984716795808, −7.87554170069646117703827877949, −7.75119734905749421249091080610, −7.17199259741541919247383076839, −7.15479985705976186082730493055, −6.01192522266309615903561756089, −5.60609386748082025149710631782, −5.56965946779049544365706938216, −5.07252470279251523848864609082, −4.44883157320725270432133685311, −3.92928183227287822437359444945, −3.16112833952143115810703523149, −3.10998675331236431894781811691, −1.91466993522813114027894941489, −1.88864307716317065101849509719, 0, 0, 1.88864307716317065101849509719, 1.91466993522813114027894941489, 3.10998675331236431894781811691, 3.16112833952143115810703523149, 3.92928183227287822437359444945, 4.44883157320725270432133685311, 5.07252470279251523848864609082, 5.56965946779049544365706938216, 5.60609386748082025149710631782, 6.01192522266309615903561756089, 7.15479985705976186082730493055, 7.17199259741541919247383076839, 7.75119734905749421249091080610, 7.87554170069646117703827877949, 8.476667599480528572984716795808, 8.998605946800609538118590352237, 9.323845620263769474202194793591, 9.525039483962604332555282034981

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