Properties

Label 4-2205e2-1.1-c3e2-0-9
Degree $4$
Conductor $4862025$
Sign $1$
Analytic cond. $16925.8$
Root an. cond. $11.4061$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·4-s + 10·5-s − 20·10-s − 66·11-s − 10·13-s − 20·16-s + 70·17-s − 140·19-s − 20·20-s + 132·22-s + 16·23-s + 75·25-s + 20·26-s + 258·29-s + 20·31-s + 200·32-s − 140·34-s + 328·37-s + 280·38-s + 300·41-s − 116·43-s + 132·44-s − 32·46-s − 30·47-s − 150·50-s + 20·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/4·4-s + 0.894·5-s − 0.632·10-s − 1.80·11-s − 0.213·13-s − 0.312·16-s + 0.998·17-s − 1.69·19-s − 0.223·20-s + 1.27·22-s + 0.145·23-s + 3/5·25-s + 0.150·26-s + 1.65·29-s + 0.115·31-s + 1.10·32-s − 0.706·34-s + 1.45·37-s + 1.19·38-s + 1.14·41-s − 0.411·43-s + 0.452·44-s − 0.102·46-s − 0.0931·47-s − 0.424·50-s + 0.0533·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4862025\)    =    \(3^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(16925.8\)
Root analytic conductor: \(11.4061\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2205} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4862025,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 6 p T + 325 p T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 10 T + 19 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 70 T + 6651 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 140 T + 14218 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 16 T + 15774 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 258 T + 40075 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 20 T + 20082 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 328 T + 106906 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 300 T + 50342 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 116 T + 160794 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 30 T + 190271 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 540 T + 212254 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 380 T + 429258 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 1080 T + 705962 T^{2} + 1080 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 468 T + 554906 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1056 T + 949550 T^{2} - 1056 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 860 T + 522934 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 2 p T - 339825 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 40 T + 703974 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 240 T - 164062 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1630 T + 2133171 T^{2} + 1630 p^{3} T^{3} + p^{6} T^{4} \)
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

−8.308804107412073013538911672436, −8.202626515721192135736894210948, −7.88079029635510478006113688897, −7.68389165186977111770916225327, −6.77087147665941103360823357633, −6.49496611055088818601213585915, −6.39496995939240137948591914470, −5.68183981694589348740438607502, −5.31536806417530739180749664975, −5.05077000555621080716121651416, −4.46054636719244234525752980371, −4.22101127843275003268718989771, −3.44822034718134925140557091536, −2.67237947422684543147802421394, −2.62638037376683885057015890499, −2.26287490741642889037184638622, −1.30568084041543222898437621453, −0.970572541258496680490855146850, 0, 0, 0.970572541258496680490855146850, 1.30568084041543222898437621453, 2.26287490741642889037184638622, 2.62638037376683885057015890499, 2.67237947422684543147802421394, 3.44822034718134925140557091536, 4.22101127843275003268718989771, 4.46054636719244234525752980371, 5.05077000555621080716121651416, 5.31536806417530739180749664975, 5.68183981694589348740438607502, 6.39496995939240137948591914470, 6.49496611055088818601213585915, 6.77087147665941103360823357633, 7.68389165186977111770916225327, 7.88079029635510478006113688897, 8.202626515721192135736894210948, 8.308804107412073013538911672436

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