Properties

Label 8-84e4-1.1-c1e4-0-0
Degree $8$
Conductor $49787136$
Sign $1$
Analytic cond. $0.202407$
Root an. cond. $0.818989$
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 − 4·3-s + 4-s + 4·6-s + 2·7-s − 3·8-s + 10·9-s − 4·12-s − 2·14-s + 16-s − 10·18-s + 12·19-s − 8·21-s + 12·24-s + 6·25-s − 20·27-s + 2·28-s − 8·29-s + 32-s + 10·36-s − 12·37-s − 12·38-s + 8·42-s − 8·47-s − 4·48-s + 6·49-s − 6·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s + 0.755·7-s − 1.06·8-s + 10/3·9-s − 1.15·12-s − 0.534·14-s + 1/4·16-s − 2.35·18-s + 2.75·19-s − 1.74·21-s + 2.44·24-s + 6/5·25-s − 3.84·27-s + 0.377·28-s − 1.48·29-s + 0.176·32-s + 5/3·36-s − 1.97·37-s − 1.94·38-s + 1.23·42-s − 1.16·47-s − 0.577·48-s + 6/7·49-s − 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.202407\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{84} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 49787136,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3024189521\)
\(L(\frac12)\) \(\approx\) \(0.3024189521\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$C_1$ \( ( 1 + T )^{4} \)
7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2 \wr C_2$ \( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 - 22 T^{2} + 682 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 82 T^{2} + 2722 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$D_{4}$ \( ( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - 102 T^{2} + 5130 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2 \wr C_2$ \( 1 - 24 T^{2} + 3774 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
61$C_2^2 \wr C_2$ \( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2 \wr C_2$ \( 1 - 144 T^{2} + 10830 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 - 114 T^{2} + 8418 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2 \wr C_2$ \( 1 - 236 T^{2} + 24310 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2 \wr C_2$ \( 1 - 288 T^{2} + 33150 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2 \wr C_2$ \( 1 - 294 T^{2} + 36618 T^{4} - 294 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.54090715318362925293792215804, −10.46670276775959189710085022577, −10.05605089236899779108555143567, −9.767264125272888931018523878461, −9.615539218424547144561418829750, −9.284772975050576703196082624885, −8.750367000463082385743015748634, −8.485088658869101834459097857355, −8.456778002023107639474112546269, −7.53555696829086494453008678384, −7.43224542851742676728711725785, −7.30083954693586896562771172570, −6.80710732998294700497151854885, −6.69264289318017612243495303152, −6.17918897770837174598230684376, −5.65037187613062237570453708006, −5.53586871994434092366007764959, −5.17481636866346656919955845781, −5.09078777665327975728685755752, −4.45884795055782048674184797584, −3.85272822630576510105703056642, −3.46753522626254578920840203411, −2.74612645231323554418663994176, −1.82160855892319050731105368928, −0.985351524338355910375931631778, 0.985351524338355910375931631778, 1.82160855892319050731105368928, 2.74612645231323554418663994176, 3.46753522626254578920840203411, 3.85272822630576510105703056642, 4.45884795055782048674184797584, 5.09078777665327975728685755752, 5.17481636866346656919955845781, 5.53586871994434092366007764959, 5.65037187613062237570453708006, 6.17918897770837174598230684376, 6.69264289318017612243495303152, 6.80710732998294700497151854885, 7.30083954693586896562771172570, 7.43224542851742676728711725785, 7.53555696829086494453008678384, 8.456778002023107639474112546269, 8.485088658869101834459097857355, 8.750367000463082385743015748634, 9.284772975050576703196082624885, 9.615539218424547144561418829750, 9.767264125272888931018523878461, 10.05605089236899779108555143567, 10.46670276775959189710085022577, 10.54090715318362925293792215804

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