Properties

Degree $4$
Conductor $680625$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 3·9-s − 4·13-s + 5·16-s − 8·19-s − 16·31-s + 9·36-s + 4·37-s − 8·43-s − 14·49-s + 12·52-s − 20·61-s − 3·64-s + 32·67-s − 28·73-s + 24·76-s + 16·79-s + 9·81-s − 20·97-s + 8·103-s − 36·109-s + 12·117-s + 121-s + 48·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 3/2·4-s − 9-s − 1.10·13-s + 5/4·16-s − 1.83·19-s − 2.87·31-s + 3/2·36-s + 0.657·37-s − 1.21·43-s − 2·49-s + 1.66·52-s − 2.56·61-s − 3/8·64-s + 3.90·67-s − 3.27·73-s + 2.75·76-s + 1.80·79-s + 81-s − 2.03·97-s + 0.788·103-s − 3.44·109-s + 1.10·117-s + 1/11·121-s + 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(680625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{680625} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 680625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
5 \( 1 \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−7.82885193609770565758085877155, −7.79601629195371107885691354892, −6.91243096690541594714752545234, −6.54918508631964327843106616026, −6.00485879019985689453618430449, −5.29765417661474514321323241134, −5.23353820210759730524176244525, −4.62665433424134304314551669114, −4.11816586584967359873614255585, −3.66351975201692451638590736882, −3.03754800212983980166528161647, −2.31804956482818418780108048277, −1.62647132327272232485132186680, 0, 0, 1.62647132327272232485132186680, 2.31804956482818418780108048277, 3.03754800212983980166528161647, 3.66351975201692451638590736882, 4.11816586584967359873614255585, 4.62665433424134304314551669114, 5.23353820210759730524176244525, 5.29765417661474514321323241134, 6.00485879019985689453618430449, 6.54918508631964327843106616026, 6.91243096690541594714752545234, 7.79601629195371107885691354892, 7.82885193609770565758085877155

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