Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{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 − 4·5-s − 9-s + 4·11-s + 16-s + 4·20-s + 11·25-s − 16·31-s + 36-s + 4·41-s − 4·44-s + 4·45-s + 10·49-s − 16·55-s − 20·59-s + 4·61-s − 64-s + 24·71-s − 4·80-s + 81-s + 20·89-s − 4·99-s − 11·100-s − 16·101-s − 20·109-s − 10·121-s + 16·124-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s − 1/3·9-s + 1.20·11-s + 1/4·16-s + 0.894·20-s + 11/5·25-s − 2.87·31-s + 1/6·36-s + 0.624·41-s − 0.603·44-s + 0.596·45-s + 10/7·49-s − 2.15·55-s − 2.60·59-s + 0.512·61-s − 1/8·64-s + 2.84·71-s − 0.447·80-s + 1/9·81-s + 2.11·89-s − 0.402·99-s − 1.09·100-s − 1.59·101-s − 1.91·109-s − 0.909·121-s + 1.43·124-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{30} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.396227$
$L(\frac12)$  $\approx$  $0.396227$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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

−17.01977398561147585569982166153, −16.96826441731702099101614986322, −16.14791334000609191876902356433, −15.63832276156840719790648214054, −14.82613947152933352288907316812, −14.60665231932445477324661417962, −13.88854108804486098115315613885, −12.96250512409104260660029756949, −12.26964569530337323975039188135, −11.95451807699588727491650213315, −11.03406984828253276966084497277, −10.79733812128479603885789976626, −9.291364285307452496699123458412, −9.062236764009944146012759381636, −8.085457671479813666280641047905, −7.48664995322902406921466101924, −6.63650189600152426497619539048, −5.35115232455633761080372105825, −4.16710137136039276355392879535, −3.56141683746843219707232773454, 3.56141683746843219707232773454, 4.16710137136039276355392879535, 5.35115232455633761080372105825, 6.63650189600152426497619539048, 7.48664995322902406921466101924, 8.085457671479813666280641047905, 9.062236764009944146012759381636, 9.291364285307452496699123458412, 10.79733812128479603885789976626, 11.03406984828253276966084497277, 11.95451807699588727491650213315, 12.26964569530337323975039188135, 12.96250512409104260660029756949, 13.88854108804486098115315613885, 14.60665231932445477324661417962, 14.82613947152933352288907316812, 15.63832276156840719790648214054, 16.14791334000609191876902356433, 16.96826441731702099101614986322, 17.01977398561147585569982166153

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