Properties

Label 8-1638e4-1.1-c1e4-0-2
Degree $8$
Conductor $7.199\times 10^{12}$
Sign $1$
Analytic cond. $29266.0$
Root an. cond. $3.61655$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 6·11-s − 4·13-s − 4·17-s + 12·25-s + 6·29-s + 6·37-s + 10·43-s − 6·44-s + 49-s − 4·52-s − 28·53-s − 24·59-s − 64-s + 24·67-s − 4·68-s + 18·71-s + 36·79-s + 12·89-s + 30·97-s + 12·100-s − 20·101-s − 52·103-s − 6·107-s + 34·113-s + 6·116-s − 121-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.80·11-s − 1.10·13-s − 0.970·17-s + 12/5·25-s + 1.11·29-s + 0.986·37-s + 1.52·43-s − 0.904·44-s + 1/7·49-s − 0.554·52-s − 3.84·53-s − 3.12·59-s − 1/8·64-s + 2.93·67-s − 0.485·68-s + 2.13·71-s + 4.05·79-s + 1.27·89-s + 3.04·97-s + 6/5·100-s − 1.99·101-s − 5.12·103-s − 0.580·107-s + 3.19·113-s + 0.557·116-s − 0.0909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(29266.0\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1638} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5528701240\)
\(L(\frac12)\) \(\approx\) \(0.5528701240\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
good5$D_4\times C_2$ \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 4 T - 19 T^{2} + 4 T^{3} + 664 T^{4} + 4 p T^{5} - 19 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \)
23$C_2^3$ \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 6 T - 19 T^{2} + 18 T^{3} + 1140 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 52 T^{2} + 1626 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 6 T + 64 T^{2} - 312 T^{3} + 1779 T^{4} - 312 p T^{5} + 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^3$ \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 10 T - 8 T^{2} - 220 T^{3} + 5515 T^{4} - 220 p T^{5} - 8 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 162 T^{2} + 10931 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
59$D_4\times C_2$ \( 1 + 24 T + 322 T^{2} + 3120 T^{3} + 24747 T^{4} + 3120 p T^{5} + 322 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 95 T^{2} + 5304 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 24 T + 370 T^{2} - 4272 T^{3} + 40059 T^{4} - 4272 p T^{5} + 370 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 18 T + 268 T^{2} - 2880 T^{3} + 28227 T^{4} - 2880 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 260 T^{2} + 27366 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 18 T + 191 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 12 T + 189 T^{2} - 1692 T^{3} + 16232 T^{4} - 1692 p T^{5} + 189 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 30 T + 520 T^{2} - 6600 T^{3} + 68091 T^{4} - 6600 p T^{5} + 520 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.81412061440473885144137882358, −6.40829989080868768616117329519, −6.28088336786786603251600828170, −6.17951509823235513325760044994, −6.04674905165356563452160751159, −5.34070015264298279430265884688, −5.23590409029145978157722946742, −5.20216383909316755572750853331, −5.05343742399378135050588325518, −4.69257207666598838490961802242, −4.59407574713558939662602346191, −4.42720917593724661239371599312, −4.11825568629710228907554964164, −3.49525555904957893082338120066, −3.48535681781101575635676774862, −3.35061895603064059981480144332, −2.87819728676688643803261904369, −2.50413742604742807317364036985, −2.45786563439892253109680955057, −2.41310867188804024984334067339, −2.13941053305896401936935349682, −1.42045467599086733677418857244, −1.20931236325008293056005258602, −0.824991909797718461465940940955, −0.14828689520451778951719055200, 0.14828689520451778951719055200, 0.824991909797718461465940940955, 1.20931236325008293056005258602, 1.42045467599086733677418857244, 2.13941053305896401936935349682, 2.41310867188804024984334067339, 2.45786563439892253109680955057, 2.50413742604742807317364036985, 2.87819728676688643803261904369, 3.35061895603064059981480144332, 3.48535681781101575635676774862, 3.49525555904957893082338120066, 4.11825568629710228907554964164, 4.42720917593724661239371599312, 4.59407574713558939662602346191, 4.69257207666598838490961802242, 5.05343742399378135050588325518, 5.20216383909316755572750853331, 5.23590409029145978157722946742, 5.34070015264298279430265884688, 6.04674905165356563452160751159, 6.17951509823235513325760044994, 6.28088336786786603251600828170, 6.40829989080868768616117329519, 6.81412061440473885144137882358

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