Properties

Label 4-1575e2-1.1-c0e2-0-0
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 − 2·7-s − 2·28-s + 2·37-s + 2·43-s + 3·49-s − 64-s − 2·67-s − 2·79-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 4-s − 2·7-s − 2·28-s + 2·37-s + 2·43-s + 3·49-s − 64-s − 2·67-s − 2·79-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 2·172-s + 173-s + 179-s + 181-s + 191-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

\(L(\frac{1}{2})\) \(\approx\) \(1.059640727\)
\(L(\frac12)\) \(\approx\) \(1.059640727\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
71$C_2^2$ \( 1 - T^{2} + T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.617057177745172190576011958428, −9.584393752283810721806893958335, −9.008455228871833970233088811475, −8.884379219407939307763741609540, −8.152954054950129406405947759117, −7.62155594495233913214084869127, −7.42001928227150077772234916988, −6.83692222167584058846613281271, −6.75890356213655413725298278223, −6.08152342081798955968313315436, −5.86429023127124861125893333481, −5.72027586380494215003391839695, −4.74124173334812975319517000054, −4.24112695776208105432444427395, −3.95235208207629412105744481474, −3.03726286108766613051093752769, −3.00540177480996256264934509273, −2.48516409891222525293330258201, −1.82560264951628206134338892523, −0.804016494520782995637871771400, 0.804016494520782995637871771400, 1.82560264951628206134338892523, 2.48516409891222525293330258201, 3.00540177480996256264934509273, 3.03726286108766613051093752769, 3.95235208207629412105744481474, 4.24112695776208105432444427395, 4.74124173334812975319517000054, 5.72027586380494215003391839695, 5.86429023127124861125893333481, 6.08152342081798955968313315436, 6.75890356213655413725298278223, 6.83692222167584058846613281271, 7.42001928227150077772234916988, 7.62155594495233913214084869127, 8.152954054950129406405947759117, 8.884379219407939307763741609540, 9.008455228871833970233088811475, 9.584393752283810721806893958335, 9.617057177745172190576011958428

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