Properties

Label 4-2475e2-1.1-c0e2-0-2
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $1.52568$
Root an. cond. $1.11138$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·11-s − 16-s − 4·89-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 2·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 2·11-s − 16-s − 4·89-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 2·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1.52568\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6125625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.402802975\)
\(L(\frac12)\) \(\approx\) \(1.402802975\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$ \( ( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−9.148943117436535000733471307183, −8.897213897461852314347870728101, −8.793860599828825712313843901958, −8.139626582578233471673799587764, −7.87279740722943014216247263140, −7.21442510577589526809801131629, −7.01917485879440151615211308252, −6.50316908836612358804794293084, −6.47194816248568494855302905191, −5.82854231331736859325744871499, −5.43950039264207885793310037888, −4.98190229141573149935883669302, −4.31087894259796945565522230331, −4.14989250529904364092095819238, −3.86423390662753462415709507789, −2.98543465640059473653356308600, −2.89133282975621820332282598464, −1.85598143194806943782795038501, −1.70428658917981024818608436446, −0.830067582061538016029864266451, 0.830067582061538016029864266451, 1.70428658917981024818608436446, 1.85598143194806943782795038501, 2.89133282975621820332282598464, 2.98543465640059473653356308600, 3.86423390662753462415709507789, 4.14989250529904364092095819238, 4.31087894259796945565522230331, 4.98190229141573149935883669302, 5.43950039264207885793310037888, 5.82854231331736859325744871499, 6.47194816248568494855302905191, 6.50316908836612358804794293084, 7.01917485879440151615211308252, 7.21442510577589526809801131629, 7.87279740722943014216247263140, 8.139626582578233471673799587764, 8.793860599828825712313843901958, 8.897213897461852314347870728101, 9.148943117436535000733471307183

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