Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \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 + 2·5-s + 4·7-s − 3·9-s − 3·16-s − 12·17-s + 2·20-s + 3·25-s + 4·28-s + 8·35-s − 3·36-s − 4·37-s − 12·41-s − 16·43-s − 6·45-s + 24·47-s + 9·49-s + 24·59-s − 12·63-s − 7·64-s + 16·67-s − 12·68-s + 16·79-s − 6·80-s + 9·81-s − 24·85-s − 12·89-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.894·5-s + 1.51·7-s − 9-s − 3/4·16-s − 2.91·17-s + 0.447·20-s + 3/5·25-s + 0.755·28-s + 1.35·35-s − 1/2·36-s − 0.657·37-s − 1.87·41-s − 2.43·43-s − 0.894·45-s + 3.50·47-s + 9/7·49-s + 3.12·59-s − 1.51·63-s − 7/8·64-s + 1.95·67-s − 1.45·68-s + 1.80·79-s − 0.670·80-s + 81-s − 2.60·85-s − 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 11025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.27642$
$L(\frac12)$  $\approx$  $1.27642$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + p T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−13.85073546011936671499247282926, −13.51046139945140927605779713303, −13.39455584922365110905441252970, −12.33787647325264601195125203552, −11.77580529432819705656530642586, −11.24803574257886029889692469881, −11.09076564889229001503688653183, −10.51196380823766213115919406936, −9.811044810689386859596554820332, −8.888675227619284864770247112347, −8.635844725493169134753519723351, −8.331081264168999936219915918912, −7.08173325171240219905876961637, −6.84319041448808681405548349722, −6.12624780347219580393116659172, −5.19569873787941261107632649192, −4.90641769224314713543468516814, −3.89188085758447586938992309253, −2.30989461230611783328334420826, −2.12350520376179601463862809320, 2.12350520376179601463862809320, 2.30989461230611783328334420826, 3.89188085758447586938992309253, 4.90641769224314713543468516814, 5.19569873787941261107632649192, 6.12624780347219580393116659172, 6.84319041448808681405548349722, 7.08173325171240219905876961637, 8.331081264168999936219915918912, 8.635844725493169134753519723351, 8.888675227619284864770247112347, 9.811044810689386859596554820332, 10.51196380823766213115919406936, 11.09076564889229001503688653183, 11.24803574257886029889692469881, 11.77580529432819705656530642586, 12.33787647325264601195125203552, 13.39455584922365110905441252970, 13.51046139945140927605779713303, 13.85073546011936671499247282926

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