Properties

Label 4-1575e2-1.1-c3e2-0-9
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $8635.61$
Root an. cond. $9.63991$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s + 14·7-s − 164·13-s − 55·16-s − 40·19-s + 42·28-s + 312·31-s − 372·37-s − 328·43-s + 147·49-s − 492·52-s + 1.58e3·61-s − 357·64-s + 88·67-s − 252·73-s − 120·76-s − 1.42e3·79-s − 2.29e3·91-s − 1.59e3·97-s + 1.83e3·103-s − 684·109-s − 770·112-s − 762·121-s + 936·124-s + 127-s + 131-s − 560·133-s + ⋯
L(s)  = 1  + 3/8·4-s + 0.755·7-s − 3.49·13-s − 0.859·16-s − 0.482·19-s + 0.283·28-s + 1.80·31-s − 1.65·37-s − 1.16·43-s + 3/7·49-s − 1.31·52-s + 3.31·61-s − 0.697·64-s + 0.160·67-s − 0.404·73-s − 0.181·76-s − 2.02·79-s − 2.64·91-s − 1.67·97-s + 1.75·103-s − 0.601·109-s − 0.649·112-s − 0.572·121-s + 0.677·124-s + 0.000698·127-s + 0.000666·131-s − 0.365·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8635.61\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2480625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 762 T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 82 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3670 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 20 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 7234 T^{2} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 10806 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 156 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 186 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 110406 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 + 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 13970 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 273130 T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 386134 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 790 T + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 44 T + p^{3} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 518146 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 + 126 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 712 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 1001450 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 - 709626 T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 798 T + p^{3} T^{2} )^{2} \)
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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

−8.708868117721363953112340269844, −8.451197787958746980682123165043, −8.131770128728905841801201329873, −7.49624444500191831164238602583, −7.28686349679267750504673946192, −6.84530971248742089321173757344, −6.72696842116784968926156911817, −6.04551346938736346731525740512, −5.23923926429886942539645962710, −5.19850248269318109320163056491, −4.77043998970589015163552518764, −4.37376762877792258587577377988, −3.88588032214105898573942609018, −3.01070291503872173611871748712, −2.60463137500452249696207702433, −2.24184616499905154903230776304, −1.86446690442122383373817726915, −1.02522391810165509156188322868, 0, 0, 1.02522391810165509156188322868, 1.86446690442122383373817726915, 2.24184616499905154903230776304, 2.60463137500452249696207702433, 3.01070291503872173611871748712, 3.88588032214105898573942609018, 4.37376762877792258587577377988, 4.77043998970589015163552518764, 5.19850248269318109320163056491, 5.23923926429886942539645962710, 6.04551346938736346731525740512, 6.72696842116784968926156911817, 6.84530971248742089321173757344, 7.28686349679267750504673946192, 7.49624444500191831164238602583, 8.131770128728905841801201329873, 8.451197787958746980682123165043, 8.708868117721363953112340269844

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