Properties

Label 4-2450e2-1.1-c1e2-0-23
Degree $4$
Conductor $6002500$
Sign $1$
Analytic cond. $382.724$
Root an. cond. $4.42304$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·3-s + 3·4-s + 8·6-s + 4·8-s + 8·9-s + 4·11-s + 12·12-s − 4·13-s + 5·16-s + 8·17-s + 16·18-s − 4·19-s + 8·22-s + 8·23-s + 16·24-s − 8·26-s + 12·27-s − 4·29-s + 6·32-s + 16·33-s + 16·34-s + 24·36-s + 4·37-s − 8·38-s − 16·39-s − 8·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 3/2·4-s + 3.26·6-s + 1.41·8-s + 8/3·9-s + 1.20·11-s + 3.46·12-s − 1.10·13-s + 5/4·16-s + 1.94·17-s + 3.77·18-s − 0.917·19-s + 1.70·22-s + 1.66·23-s + 3.26·24-s − 1.56·26-s + 2.30·27-s − 0.742·29-s + 1.06·32-s + 2.78·33-s + 2.74·34-s + 4·36-s + 0.657·37-s − 1.29·38-s − 2.56·39-s − 1.24·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6002500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(382.724\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6002500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(19.87885117\)
\(L(\frac12)\) \(\approx\) \(19.87885117\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 20 T + 216 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 166 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 152 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
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

−9.005542285351456541315725691220, −8.856846334554428020864098635980, −8.278826285656810990382060867921, −7.896593738074211233654737460013, −7.63035902658805148773042849001, −7.21490593014409896443050221316, −6.91562426009550679172609142242, −6.59610180173220777972555097635, −5.73493691343491563302375784385, −5.73458901830460254479162418533, −5.07033450102132200345156853276, −4.58709144064465782443475246714, −4.09779435192815423196564673175, −3.87612202042714297345548299506, −3.41146104057933322157267710421, −2.93643414327279619723543248357, −2.62896029032547538144926146264, −2.35040371418758023045380258684, −1.50993144313136663672942612511, −1.13095744429079864449160595150, 1.13095744429079864449160595150, 1.50993144313136663672942612511, 2.35040371418758023045380258684, 2.62896029032547538144926146264, 2.93643414327279619723543248357, 3.41146104057933322157267710421, 3.87612202042714297345548299506, 4.09779435192815423196564673175, 4.58709144064465782443475246714, 5.07033450102132200345156853276, 5.73458901830460254479162418533, 5.73493691343491563302375784385, 6.59610180173220777972555097635, 6.91562426009550679172609142242, 7.21490593014409896443050221316, 7.63035902658805148773042849001, 7.896593738074211233654737460013, 8.278826285656810990382060867921, 8.856846334554428020864098635980, 9.005542285351456541315725691220

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