Properties

Label 4-1575e2-1.1-c1e2-0-10
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $158.166$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·7-s − 4·13-s − 3·16-s − 8·19-s + 2·28-s − 8·31-s − 4·37-s + 8·43-s + 3·49-s + 4·52-s − 20·61-s + 7·64-s + 8·67-s − 28·73-s + 8·76-s + 16·79-s + 8·91-s − 28·97-s + 8·103-s + 4·109-s + 6·112-s − 10·121-s + 8·124-s + 127-s + 131-s + 16·133-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.755·7-s − 1.10·13-s − 3/4·16-s − 1.83·19-s + 0.377·28-s − 1.43·31-s − 0.657·37-s + 1.21·43-s + 3/7·49-s + 0.554·52-s − 2.56·61-s + 7/8·64-s + 0.977·67-s − 3.27·73-s + 0.917·76-s + 1.80·79-s + 0.838·91-s − 2.84·97-s + 0.788·103-s + 0.383·109-s + 0.566·112-s − 0.909·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(158.166\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−9.152802080925025956854116387692, −9.041739330556870645526973791472, −8.486047191146135179383478045838, −8.138830421689102299251651732368, −7.48318071343520741736810069807, −7.26534813260720064108662301111, −6.87853029702903180920342868933, −6.32411000044886643798810545930, −6.05420919147302561108896002088, −5.55949584772863814681134226966, −4.91222842779235652798525196373, −4.72453124273459494740236995495, −3.97601938448959508043292558696, −3.95821704618757537757290215994, −3.11353433074374222256109561769, −2.55220020995291457056915011569, −2.17644355062192818414262405510, −1.41141467297432757952072339356, 0, 0, 1.41141467297432757952072339356, 2.17644355062192818414262405510, 2.55220020995291457056915011569, 3.11353433074374222256109561769, 3.95821704618757537757290215994, 3.97601938448959508043292558696, 4.72453124273459494740236995495, 4.91222842779235652798525196373, 5.55949584772863814681134226966, 6.05420919147302561108896002088, 6.32411000044886643798810545930, 6.87853029702903180920342868933, 7.26534813260720064108662301111, 7.48318071343520741736810069807, 8.138830421689102299251651732368, 8.486047191146135179383478045838, 9.041739330556870645526973791472, 9.152802080925025956854116387692

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