Properties

Label 4-585e2-1.1-c1e2-0-5
Degree $4$
Conductor $342225$
Sign $1$
Analytic cond. $21.8205$
Root an. cond. $2.16130$
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-s − 2·5-s + 4·7-s − 4·10-s − 4·11-s − 2·13-s + 8·14-s + 16-s + 4·17-s + 4·19-s − 2·20-s − 8·22-s + 3·25-s − 4·26-s + 4·28-s + 12·31-s − 2·32-s + 8·34-s − 8·35-s + 8·38-s + 12·41-s − 8·43-s − 4·44-s + 4·47-s + 6·49-s + 6·50-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s − 1.26·10-s − 1.20·11-s − 0.554·13-s + 2.13·14-s + 1/4·16-s + 0.970·17-s + 0.917·19-s − 0.447·20-s − 1.70·22-s + 3/5·25-s − 0.784·26-s + 0.755·28-s + 2.15·31-s − 0.353·32-s + 1.37·34-s − 1.35·35-s + 1.29·38-s + 1.87·41-s − 1.21·43-s − 0.603·44-s + 0.583·47-s + 6/7·49-s + 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(342225\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(21.8205\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{585} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 342225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.441391519\)
\(L(\frac12)\) \(\approx\) \(3.441391519\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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

−11.02327904860097487853365600695, −10.49217856516854857244707613659, −10.36110934961180151067116332977, −9.714286165109304329331025621075, −9.094636004158736703950196942447, −8.601144137217098123137355612783, −7.931217069701776126282701574024, −7.82748707143792572786159821091, −7.52806058553861407863894042151, −7.00274043662204370004099982388, −6.06463888489816548881869814765, −5.70839080855422604232435173677, −5.06077252307295835194499274723, −4.89361651758623846143271627664, −4.42392998264889679461814329749, −4.08089846690771216958976744620, −3.07921870702749460485365586304, −2.97450618779523508374006928154, −1.91204956050814003374058696388, −0.897256041534199990641944032128, 0.897256041534199990641944032128, 1.91204956050814003374058696388, 2.97450618779523508374006928154, 3.07921870702749460485365586304, 4.08089846690771216958976744620, 4.42392998264889679461814329749, 4.89361651758623846143271627664, 5.06077252307295835194499274723, 5.70839080855422604232435173677, 6.06463888489816548881869814765, 7.00274043662204370004099982388, 7.52806058553861407863894042151, 7.82748707143792572786159821091, 7.931217069701776126282701574024, 8.601144137217098123137355612783, 9.094636004158736703950196942447, 9.714286165109304329331025621075, 10.36110934961180151067116332977, 10.49217856516854857244707613659, 11.02327904860097487853365600695

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