Properties

Degree $4$
Conductor $2480625$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s + 3·19-s − 28-s + 37-s + 4·43-s − 3·61-s − 64-s − 67-s − 3·73-s + 3·76-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯
L(s)  = 1  + 4-s − 7-s + 3·19-s − 28-s + 37-s + 4·43-s − 3·61-s − 64-s − 67-s − 3·73-s + 3·76-s + 79-s − 3·103-s − 109-s + 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.410286689\)
\(L(\frac12)\) \(\approx\) \(1.410286689\)
\(L(1)\) 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 \( 1 \)
7$C_2$ \( 1 + T + T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$ \( ( 1 - T )^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−9.571165917370971600287195208813, −9.541884241741345333064276369543, −9.106073597725074568894866773615, −8.893357507536551178093015173587, −7.87874864765669193479236167158, −7.79488126140241289617004408679, −7.37919937047148341501953087265, −7.10070670875801345571919352382, −6.69253119144104813150899422309, −5.98100840600319990675127526828, −5.85900599019417237478912698898, −5.63389040558050400263991941961, −4.76475524543023702655972934511, −4.41956878938508141687838840019, −3.82677478317679182518084185679, −3.12760345105075596315641598121, −2.88104399043898628402685523472, −2.59511306074975866219956695347, −1.60655495480169230664533452451, −1.02460460141546533731090149360, 1.02460460141546533731090149360, 1.60655495480169230664533452451, 2.59511306074975866219956695347, 2.88104399043898628402685523472, 3.12760345105075596315641598121, 3.82677478317679182518084185679, 4.41956878938508141687838840019, 4.76475524543023702655972934511, 5.63389040558050400263991941961, 5.85900599019417237478912698898, 5.98100840600319990675127526828, 6.69253119144104813150899422309, 7.10070670875801345571919352382, 7.37919937047148341501953087265, 7.79488126140241289617004408679, 7.87874864765669193479236167158, 8.893357507536551178093015173587, 9.106073597725074568894866773615, 9.541884241741345333064276369543, 9.571165917370971600287195208813

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