Properties

Label 4-1575e2-1.1-c0e2-0-3
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $0.617839$
Root an. cond. $0.886581$
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  + 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: $1$
Analytic conductor: \(0.617839\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.749818988\)
\(L(\frac12)\) \(\approx\) \(1.749818988\)
\(L(1)\) 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

−9.759633078010087114845882513777, −9.565360558033900248364380465734, −8.909757361405363848103438008072, −8.715781220519617269067992459067, −8.010216691182826770194000175682, −7.68824493082618607930019820666, −7.65024385460216583404467428725, −6.99329281199656667288320251889, −6.51519360316686909037643447562, −6.47562218671681984953798436746, −5.60313709408012586111736879051, −5.25775626116668737001177135895, −4.91471620087078538433950296498, −4.66840421544948942162143250317, −3.56203413978644367886573327063, −3.43807187385475128884232854567, −2.96215741090627392784972702705, −2.16877986192885348746052775847, −1.68604251927303734649373186138, −1.20230275749537431270299909491, 1.20230275749537431270299909491, 1.68604251927303734649373186138, 2.16877986192885348746052775847, 2.96215741090627392784972702705, 3.43807187385475128884232854567, 3.56203413978644367886573327063, 4.66840421544948942162143250317, 4.91471620087078538433950296498, 5.25775626116668737001177135895, 5.60313709408012586111736879051, 6.47562218671681984953798436746, 6.51519360316686909037643447562, 6.99329281199656667288320251889, 7.65024385460216583404467428725, 7.68824493082618607930019820666, 8.010216691182826770194000175682, 8.715781220519617269067992459067, 8.909757361405363848103438008072, 9.565360558033900248364380465734, 9.759633078010087114845882513777

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