Properties

Degree $8$
Conductor $1.216\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s − 6·7-s + 2·9-s − 4·12-s + 3·16-s − 12·17-s − 12·21-s + 6·27-s + 12·28-s − 4·36-s − 12·37-s + 8·41-s − 16·43-s + 8·47-s + 6·48-s + 18·49-s − 24·51-s − 12·63-s − 4·64-s + 48·67-s + 24·68-s + 11·81-s + 4·83-s + 24·84-s − 40·89-s − 32·101-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s − 2.26·7-s + 2/3·9-s − 1.15·12-s + 3/4·16-s − 2.91·17-s − 2.61·21-s + 1.15·27-s + 2.26·28-s − 2/3·36-s − 1.97·37-s + 1.24·41-s − 2.43·43-s + 1.16·47-s + 0.866·48-s + 18/7·49-s − 3.36·51-s − 1.51·63-s − 1/2·64-s + 5.86·67-s + 2.91·68-s + 11/9·81-s + 0.439·83-s + 2.61·84-s − 4.23·89-s − 3.18·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.184783981\)
\(L(\frac12)\) \(\approx\) \(1.184783981\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_4\times C_2$ \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 184 T^{2} + 15406 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 2 T - 78 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.87171882124857471827774169151, −6.86115400907298562547086373970, −6.81783940051827124334884273252, −6.80948714588872995750633334259, −6.33133131518256237060506486834, −5.89935941887595890932446585988, −5.68556478748470897119337239977, −5.66133238956742418883971715771, −5.13928255959841595481325715070, −5.09411750310184589743828675343, −4.69416515476614367899637449225, −4.48036056640841011429084416583, −4.20956438804436174170426597315, −3.95604944487530147394437667590, −3.70614113597681864478282335179, −3.60181022102132425019819931696, −3.33776836665511901497326538238, −2.89100182024569108966458522074, −2.63752327217385364092547780433, −2.59889782380549954575825303729, −2.16756668792110497854861664515, −1.80268033401864132652597302012, −1.40699466418107158754766916260, −0.58126049432332938940186507270, −0.35789036118455111709010326276, 0.35789036118455111709010326276, 0.58126049432332938940186507270, 1.40699466418107158754766916260, 1.80268033401864132652597302012, 2.16756668792110497854861664515, 2.59889782380549954575825303729, 2.63752327217385364092547780433, 2.89100182024569108966458522074, 3.33776836665511901497326538238, 3.60181022102132425019819931696, 3.70614113597681864478282335179, 3.95604944487530147394437667590, 4.20956438804436174170426597315, 4.48036056640841011429084416583, 4.69416515476614367899637449225, 5.09411750310184589743828675343, 5.13928255959841595481325715070, 5.66133238956742418883971715771, 5.68556478748470897119337239977, 5.89935941887595890932446585988, 6.33133131518256237060506486834, 6.80948714588872995750633334259, 6.81783940051827124334884273252, 6.86115400907298562547086373970, 6.87171882124857471827774169151

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