Properties

Label 4-2541e2-1.1-c1e2-0-9
Degree $4$
Conductor $6456681$
Sign $1$
Analytic cond. $411.683$
Root an. cond. $4.50444$
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-s + 2·3-s − 2·4-s − 4·5-s + 2·6-s − 2·7-s − 3·8-s + 3·9-s − 4·10-s − 4·12-s + 8·13-s − 2·14-s − 8·15-s + 16-s + 3·18-s + 6·19-s + 8·20-s − 4·21-s + 8·23-s − 6·24-s + 7·25-s + 8·26-s + 4·27-s + 4·28-s − 6·29-s − 8·30-s + 2·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 4-s − 1.78·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 9-s − 1.26·10-s − 1.15·12-s + 2.21·13-s − 0.534·14-s − 2.06·15-s + 1/4·16-s + 0.707·18-s + 1.37·19-s + 1.78·20-s − 0.872·21-s + 1.66·23-s − 1.22·24-s + 7/5·25-s + 1.56·26-s + 0.769·27-s + 0.755·28-s − 1.11·29-s − 1.46·30-s + 0.353·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6456681\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(411.683\)
Root analytic conductor: \(4.50444\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2541} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6456681,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.532065508\)
\(L(\frac12)\) \(\approx\) \(3.532065508\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 20 T + 241 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−8.988114701860162842964666795498, −8.857143704386704643850720887222, −8.170794739156512386351255390664, −8.091215445843681269208863595386, −7.75286598807876883330113328563, −7.24857412388667026438192650925, −6.74894279595652241945681546692, −6.66498985060980633707361761410, −5.69993419467715303090900653717, −5.64248568446366900615917976346, −5.00709556257650255379608548006, −4.58447984416195775358413899347, −4.02555076456381273504056017435, −3.82500538898727930438318084747, −3.43870567717536772254314128943, −3.39825446398542176220309226827, −2.79248104188882076909598089751, −1.98827768583556639330875158522, −0.821788043614078500906236176220, −0.814502883612581387160443608813, 0.814502883612581387160443608813, 0.821788043614078500906236176220, 1.98827768583556639330875158522, 2.79248104188882076909598089751, 3.39825446398542176220309226827, 3.43870567717536772254314128943, 3.82500538898727930438318084747, 4.02555076456381273504056017435, 4.58447984416195775358413899347, 5.00709556257650255379608548006, 5.64248568446366900615917976346, 5.69993419467715303090900653717, 6.66498985060980633707361761410, 6.74894279595652241945681546692, 7.24857412388667026438192650925, 7.75286598807876883330113328563, 8.091215445843681269208863595386, 8.170794739156512386351255390664, 8.857143704386704643850720887222, 8.988114701860162842964666795498

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