Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{4} $
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  − 3·2-s + 3·3-s + 4·4-s + 5-s − 9·6-s − 3·8-s + 6·9-s − 3·10-s − 6·11-s + 12·12-s + 3·15-s + 3·16-s − 6·17-s − 18·18-s − 6·19-s + 4·20-s + 18·22-s − 6·23-s − 9·24-s + 9·27-s − 9·30-s − 6·31-s − 6·32-s − 18·33-s + 18·34-s + 24·36-s + 2·37-s + ⋯
L(s)  = 1  − 2.12·2-s + 1.73·3-s + 2·4-s + 0.447·5-s − 3.67·6-s − 1.06·8-s + 2·9-s − 0.948·10-s − 1.80·11-s + 3.46·12-s + 0.774·15-s + 3/4·16-s − 1.45·17-s − 4.24·18-s − 1.37·19-s + 0.894·20-s + 3.83·22-s − 1.25·23-s − 1.83·24-s + 1.73·27-s − 1.64·30-s − 1.07·31-s − 1.06·32-s − 3.13·33-s + 3.08·34-s + 4·36-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(540225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{735} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 540225,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.618682\)
\(L(\frac12)\)  \(\approx\)  \(0.618682\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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

−10.48861415732409790631254870289, −9.809771940038829417850968050782, −9.806687907796994709560794329683, −9.125123988989970037671485714820, −8.912474035453363200345648395802, −8.454232592329778857520417744689, −8.293573124698890403220357375671, −7.70699356168811298859583603405, −7.69927729188480473622791704470, −6.82708038342572479063033555453, −6.69400220262105208036925183015, −5.69691142222737311803911751904, −5.43719121558303435854774409714, −4.28606446527974744721778313483, −4.24135233994669874278140405937, −3.27579332436581888490403557660, −2.60677393596056049559788080092, −2.07819260618057647799877182396, −1.90854414996012034068781250975, −0.49141837750385909231436555336, 0.49141837750385909231436555336, 1.90854414996012034068781250975, 2.07819260618057647799877182396, 2.60677393596056049559788080092, 3.27579332436581888490403557660, 4.24135233994669874278140405937, 4.28606446527974744721778313483, 5.43719121558303435854774409714, 5.69691142222737311803911751904, 6.69400220262105208036925183015, 6.82708038342572479063033555453, 7.69927729188480473622791704470, 7.70699356168811298859583603405, 8.293573124698890403220357375671, 8.454232592329778857520417744689, 8.912474035453363200345648395802, 9.125123988989970037671485714820, 9.806687907796994709560794329683, 9.809771940038829417850968050782, 10.48861415732409790631254870289

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