Properties

Degree $4$
Conductor $1125$
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  − 3·4-s + 5-s + 9-s − 8·11-s + 5·16-s + 8·19-s − 3·20-s + 25-s − 4·29-s − 3·36-s + 20·41-s + 24·44-s + 45-s − 14·49-s − 8·55-s − 8·59-s − 4·61-s − 3·64-s − 16·71-s − 24·76-s + 5·80-s + 81-s − 12·89-s + 8·95-s − 8·99-s − 3·100-s + 12·101-s + ⋯
L(s)  = 1  − 3/2·4-s + 0.447·5-s + 1/3·9-s − 2.41·11-s + 5/4·16-s + 1.83·19-s − 0.670·20-s + 1/5·25-s − 0.742·29-s − 1/2·36-s + 3.12·41-s + 3.61·44-s + 0.149·45-s − 2·49-s − 1.07·55-s − 1.04·59-s − 0.512·61-s − 3/8·64-s − 1.89·71-s − 2.75·76-s + 0.559·80-s + 1/9·81-s − 1.27·89-s + 0.820·95-s − 0.804·99-s − 0.299·100-s + 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1125\)    =    \(3^{2} \cdot 5^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1125} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1125,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4386469691\)
\(L(\frac12)\) \(\approx\) \(0.4386469691\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.00692585049802430183598730465, −13.37224542302974911325065415564, −12.89741011334273074809178538613, −12.64617876135106218873747419153, −11.42796116649064334018778191223, −10.67892245123374144028858824682, −9.965958896202437987484594622667, −9.523451675812284265792380668213, −8.843956595114680827452383410006, −7.68003844385765985083943135134, −7.66488013441745380243230842523, −5.86497653842965752491328535183, −5.23920392624592057055772361749, −4.52160444884874669934496061646, −3.01549784140686353642765162463, 3.01549784140686353642765162463, 4.52160444884874669934496061646, 5.23920392624592057055772361749, 5.86497653842965752491328535183, 7.66488013441745380243230842523, 7.68003844385765985083943135134, 8.843956595114680827452383410006, 9.523451675812284265792380668213, 9.965958896202437987484594622667, 10.67892245123374144028858824682, 11.42796116649064334018778191223, 12.64617876135106218873747419153, 12.89741011334273074809178538613, 13.37224542302974911325065415564, 14.00692585049802430183598730465

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