Properties

Label 4-230e2-1.1-c1e2-0-2
Degree $4$
Conductor $52900$
Sign $1$
Analytic cond. $3.37294$
Root an. cond. $1.35519$
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 + 3·3-s + 3·4-s + 2·5-s − 6·6-s + 3·7-s − 4·8-s + 4·9-s − 4·10-s − 7·11-s + 9·12-s + 3·13-s − 6·14-s + 6·15-s + 5·16-s + 3·17-s − 8·18-s + 19-s + 6·20-s + 9·21-s + 14·22-s − 2·23-s − 12·24-s + 3·25-s − 6·26-s + 6·27-s + 9·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.894·5-s − 2.44·6-s + 1.13·7-s − 1.41·8-s + 4/3·9-s − 1.26·10-s − 2.11·11-s + 2.59·12-s + 0.832·13-s − 1.60·14-s + 1.54·15-s + 5/4·16-s + 0.727·17-s − 1.88·18-s + 0.229·19-s + 1.34·20-s + 1.96·21-s + 2.98·22-s − 0.417·23-s − 2.44·24-s + 3/5·25-s − 1.17·26-s + 1.15·27-s + 1.70·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(52900\)    =    \(2^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3.37294\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{230} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 52900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.598771131\)
\(L(\frac12)\) \(\approx\) \(1.598771131\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 29 T + 349 T^{2} + 29 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 9 T + 211 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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

−12.54562329491617748097097436723, −11.75744974693688806614818154203, −11.28144094921353663359496219197, −10.71046230091007479456466608837, −10.29941953718408687215638141424, −10.06568592678611651751133565890, −9.258305412585825402855487750376, −9.132013969731634669302277945894, −8.341857495284555451505842873076, −8.194436290827410356788454580987, −7.67615958829551750295562810625, −7.57180952442899203360257444861, −6.49361484218236037802429356517, −5.91211893904072091645324921101, −5.22867063412821604665322838870, −4.55549946053216157588369309046, −3.18920512476946472385632829776, −2.91208024671020648339608776976, −2.11808372085269199478702255532, −1.46438094505397839462929047850, 1.46438094505397839462929047850, 2.11808372085269199478702255532, 2.91208024671020648339608776976, 3.18920512476946472385632829776, 4.55549946053216157588369309046, 5.22867063412821604665322838870, 5.91211893904072091645324921101, 6.49361484218236037802429356517, 7.57180952442899203360257444861, 7.67615958829551750295562810625, 8.194436290827410356788454580987, 8.341857495284555451505842873076, 9.132013969731634669302277945894, 9.258305412585825402855487750376, 10.06568592678611651751133565890, 10.29941953718408687215638141424, 10.71046230091007479456466608837, 11.28144094921353663359496219197, 11.75744974693688806614818154203, 12.54562329491617748097097436723

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