Properties

Label 4-857500-1.1-c1e2-0-3
Degree $4$
Conductor $857500$
Sign $1$
Analytic cond. $54.6749$
Root an. cond. $2.71923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 7-s − 4·8-s − 6·9-s + 8·11-s − 2·14-s + 5·16-s + 12·18-s − 16·22-s + 3·28-s + 12·29-s − 6·32-s − 18·36-s + 20·37-s − 8·43-s + 24·44-s + 49-s + 4·53-s − 4·56-s − 24·58-s − 6·63-s + 7·64-s + 24·67-s − 32·71-s + 24·72-s − 40·74-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 2·9-s + 2.41·11-s − 0.534·14-s + 5/4·16-s + 2.82·18-s − 3.41·22-s + 0.566·28-s + 2.22·29-s − 1.06·32-s − 3·36-s + 3.28·37-s − 1.21·43-s + 3.61·44-s + 1/7·49-s + 0.549·53-s − 0.534·56-s − 3.15·58-s − 0.755·63-s + 7/8·64-s + 2.93·67-s − 3.79·71-s + 2.82·72-s − 4.64·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(857500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(54.6749\)
Root analytic conductor: \(2.71923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{857500} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 857500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.118759399\)
\(L(\frac12)\) \(\approx\) \(1.118759399\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
7$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.301913068586907274040785472708, −8.089729535357517933440155073370, −7.42472757379966149042918376603, −6.82123838368350264589622397869, −6.56718920518778472538358027386, −6.01430128602422585899125610915, −5.91548647651224347146838875772, −5.14037714155681614978134696209, −4.36823223624486251008730105784, −4.03694373686491315845893969603, −2.98711940319117381089884417433, −2.95779572995708486306206498227, −2.11286590149397096544933120998, −1.27008084899723889248060652436, −0.71479745199007503910946566518, 0.71479745199007503910946566518, 1.27008084899723889248060652436, 2.11286590149397096544933120998, 2.95779572995708486306206498227, 2.98711940319117381089884417433, 4.03694373686491315845893969603, 4.36823223624486251008730105784, 5.14037714155681614978134696209, 5.91548647651224347146838875772, 6.01430128602422585899125610915, 6.56718920518778472538358027386, 6.82123838368350264589622397869, 7.42472757379966149042918376603, 8.089729535357517933440155073370, 8.301913068586907274040785472708

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