Properties

Label 8-1170e4-1.1-c1e4-0-22
Degree $8$
Conductor $1.874\times 10^{12}$
Sign $1$
Analytic cond. $7618.19$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 6·11-s + 8·17-s + 6·19-s − 2·25-s − 4·29-s − 24·37-s − 12·43-s + 6·44-s − 5·49-s + 8·53-s + 12·59-s + 8·61-s − 64-s + 12·67-s + 8·68-s + 24·71-s + 6·76-s + 40·79-s + 42·89-s + 36·97-s − 2·100-s + 4·101-s + 16·103-s − 12·107-s − 24·113-s − 4·116-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.80·11-s + 1.94·17-s + 1.37·19-s − 2/5·25-s − 0.742·29-s − 3.94·37-s − 1.82·43-s + 0.904·44-s − 5/7·49-s + 1.09·53-s + 1.56·59-s + 1.02·61-s − 1/8·64-s + 1.46·67-s + 0.970·68-s + 2.84·71-s + 0.688·76-s + 4.50·79-s + 4.45·89-s + 3.65·97-s − 1/5·100-s + 0.398·101-s + 1.57·103-s − 1.16·107-s − 2.25·113-s − 0.371·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(7618.19\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1170} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.991090696\)
\(L(\frac12)\) \(\approx\) \(5.991090696\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3 \( 1 \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
good7$C_2^3$ \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 6 T + 3 p T^{2} - 126 T^{3} + 452 T^{4} - 126 p T^{5} + 3 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 6 T + 37 T^{2} - 150 T^{3} + 492 T^{4} - 150 p T^{5} + 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 4 T - 34 T^{2} - 32 T^{3} + 1195 T^{4} - 32 p T^{5} - 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 20 T^{2} + 1254 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 24 T + 313 T^{2} + 2904 T^{3} + 20376 T^{4} + 2904 p T^{5} + 313 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^3$ \( 1 + 66 T^{2} + 2675 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 146 T^{2} + 9315 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 4 T + 107 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 12 T + 114 T^{2} - 792 T^{3} + 3707 T^{4} - 792 p T^{5} + 114 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 8 T - 62 T^{2} - 32 T^{3} + 8251 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 12 T + 190 T^{2} - 1704 T^{3} + 18891 T^{4} - 1704 p T^{5} + 190 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 24 T + 346 T^{2} - 3696 T^{3} + 32307 T^{4} - 3696 p T^{5} + 346 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 236 T^{2} + 25974 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 42 T + 909 T^{2} - 13482 T^{3} + 147452 T^{4} - 13482 p T^{5} + 909 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 36 T + 730 T^{2} - 10728 T^{3} + 121299 T^{4} - 10728 p T^{5} + 730 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.71098240277930954339833224241, −6.69740109799727226498089585067, −6.62447042026697429664124930069, −6.54924859775625570488700779780, −6.27131657792933937087280123357, −5.89985703847032863474152365346, −5.48215738354254951062616620048, −5.26013810330017264984537899787, −5.22028760132195090630312365516, −5.15261379048501728884108604007, −5.01174873928790101506290426554, −4.43760280876518864695344612559, −4.10509633565793349291483727393, −3.73511570805105454268350055854, −3.56620769204012685251646693123, −3.42349608106645822069929463647, −3.41238740382961156422863525069, −3.26499436897199950569818017737, −2.46473093580853033710750156575, −2.09264818778133787843001754403, −1.95699256349154985658221363673, −1.89377481129216352097404021403, −1.13197727366347250905797577908, −0.989877565214531816736208764081, −0.58821105687665937743756734373, 0.58821105687665937743756734373, 0.989877565214531816736208764081, 1.13197727366347250905797577908, 1.89377481129216352097404021403, 1.95699256349154985658221363673, 2.09264818778133787843001754403, 2.46473093580853033710750156575, 3.26499436897199950569818017737, 3.41238740382961156422863525069, 3.42349608106645822069929463647, 3.56620769204012685251646693123, 3.73511570805105454268350055854, 4.10509633565793349291483727393, 4.43760280876518864695344612559, 5.01174873928790101506290426554, 5.15261379048501728884108604007, 5.22028760132195090630312365516, 5.26013810330017264984537899787, 5.48215738354254951062616620048, 5.89985703847032863474152365346, 6.27131657792933937087280123357, 6.54924859775625570488700779780, 6.62447042026697429664124930069, 6.69740109799727226498089585067, 6.71098240277930954339833224241

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