Properties

Label 4-1620e2-1.1-c3e2-0-1
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $9136.12$
Root an. cond. $9.77666$
Motivic weight $3$
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  + 5·5-s + 22·7-s + 9·11-s − 17·13-s − 150·17-s − 8·19-s − 183·23-s − 129·29-s + 187·31-s + 110·35-s − 68·37-s − 264·41-s − 443·43-s − 609·47-s + 343·49-s − 456·53-s + 45·55-s − 60·59-s + 454·61-s − 85·65-s + 244·67-s + 888·71-s + 796·73-s + 198·77-s + 349·79-s − 1.03e3·83-s − 750·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.18·7-s + 0.246·11-s − 0.362·13-s − 2.14·17-s − 0.0965·19-s − 1.65·23-s − 0.826·29-s + 1.08·31-s + 0.531·35-s − 0.302·37-s − 1.00·41-s − 1.57·43-s − 1.89·47-s + 49-s − 1.18·53-s + 0.110·55-s − 0.132·59-s + 0.952·61-s − 0.162·65-s + 0.444·67-s + 1.48·71-s + 1.27·73-s + 0.293·77-s + 0.497·79-s − 1.37·83-s − 0.957·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(9136.12\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1369449062\)
\(L(\frac12)\) \(\approx\) \(0.1369449062\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - p T + p^{2} T^{2} \)
good7$C_2^2$ \( 1 - 22 T + 141 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 9 T - 1250 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 17 T - 1908 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 75 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 183 T + 21322 T^{2} + 183 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 129 T - 7748 T^{2} + 129 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 187 T + 5178 T^{2} - 187 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 264 T + 775 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 443 T + 116742 T^{2} + 443 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 609 T + 267058 T^{2} + 609 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 228 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 60 T - 201779 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 454 T - 20865 T^{2} - 454 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 244 T - 241227 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 444 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 398 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 349 T - 371238 T^{2} - 349 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 1038 T + 505657 T^{2} + 1038 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 852 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 914 T - 77277 T^{2} + 914 p^{3} T^{3} + p^{6} T^{4} \)
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

−9.385622168920437679731523821848, −8.655668778363272125702349603063, −8.397628126050123129068039757861, −8.141514862239300309393068210237, −7.83214622170171280007631056117, −7.12166496940863698586099927593, −6.78804189495378307992983748081, −6.38478529071942104113336604242, −6.17655481834346146560559062648, −5.41339706787110531867111322979, −5.07012624363549980551679499259, −4.66928225430683723737479724075, −4.40482675211002225245496698949, −3.68242256193953417426220752352, −3.41541412313086067282456607700, −2.30421208097459375534530624418, −2.26474847239114265376531246039, −1.73658514844953754449671948837, −1.19245964081166023620335598029, −0.07705602809717099220043430140, 0.07705602809717099220043430140, 1.19245964081166023620335598029, 1.73658514844953754449671948837, 2.26474847239114265376531246039, 2.30421208097459375534530624418, 3.41541412313086067282456607700, 3.68242256193953417426220752352, 4.40482675211002225245496698949, 4.66928225430683723737479724075, 5.07012624363549980551679499259, 5.41339706787110531867111322979, 6.17655481834346146560559062648, 6.38478529071942104113336604242, 6.78804189495378307992983748081, 7.12166496940863698586099927593, 7.83214622170171280007631056117, 8.141514862239300309393068210237, 8.397628126050123129068039757861, 8.655668778363272125702349603063, 9.385622168920437679731523821848

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