Properties

Degree $8$
Conductor $1.945\times 10^{13}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·7-s + 9-s − 20·13-s + 2·19-s + 4·21-s + 12·23-s − 2·27-s + 8·31-s + 10·37-s − 40·39-s − 24·41-s + 16·43-s + 12·47-s + 7·49-s + 4·57-s − 12·59-s + 2·61-s + 2·63-s − 2·67-s + 24·69-s + 10·73-s − 10·79-s − 4·81-s − 24·83-s − 24·89-s − 40·91-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1/3·9-s − 5.54·13-s + 0.458·19-s + 0.872·21-s + 2.50·23-s − 0.384·27-s + 1.43·31-s + 1.64·37-s − 6.40·39-s − 3.74·41-s + 2.43·43-s + 1.75·47-s + 49-s + 0.529·57-s − 1.56·59-s + 0.256·61-s + 0.251·63-s − 0.244·67-s + 2.88·69-s + 1.17·73-s − 1.12·79-s − 4/9·81-s − 2.63·83-s − 2.54·89-s − 4.19·91-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2100} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6044333471\)
\(L(\frac12)\) \(\approx\) \(0.6044333471\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 - T + T^{2} )^{2} \)
5 \( 1 \)
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
good11$C_2^3$ \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
17$C_2^3$ \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 2 T - 17 T^{2} + 34 T^{3} + 4 T^{4} + 34 p T^{5} - 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( ( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 12 T + 32 T^{2} - 216 T^{3} + 3567 T^{4} - 216 p T^{5} + 32 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
67$D_4\times C_2$ \( 1 + 2 T + 31 T^{2} - 322 T^{3} - 4028 T^{4} - 322 p T^{5} + 31 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 10 T + T^{2} + 470 T^{3} - 2828 T^{4} + 470 p T^{5} + p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 70 p T^{5} - 65 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 12 T + 184 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 24 T + 272 T^{2} + 3024 T^{3} + 33231 T^{4} + 3024 p T^{5} + 272 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.59018359807203580378296515854, −6.43963969332001296391871019668, −5.90786669397225819413715186453, −5.64082083506837630840228247212, −5.63900081481304789680064521445, −5.37628810429703844266730570057, −5.02425844198564675425156190050, −4.99008084440527126610594509979, −4.76123984050685927758013431927, −4.52246026204136968592771709365, −4.44611033473386652078181611597, −4.32749048261804093995022526424, −3.85117540421915734118539905393, −3.59175019709731634243443303367, −3.08068306119921051674535457002, −3.02716093617368448069537749681, −2.74176656695706000903946859414, −2.72058292628786625949453293373, −2.38599113892756873831395510877, −2.27130608841720547962431179718, −2.03586962395108747571328974402, −1.46774573674487514944931020117, −1.18145058467931711688670654678, −0.78422482273165151599192339609, −0.12347801075692190203091457018, 0.12347801075692190203091457018, 0.78422482273165151599192339609, 1.18145058467931711688670654678, 1.46774573674487514944931020117, 2.03586962395108747571328974402, 2.27130608841720547962431179718, 2.38599113892756873831395510877, 2.72058292628786625949453293373, 2.74176656695706000903946859414, 3.02716093617368448069537749681, 3.08068306119921051674535457002, 3.59175019709731634243443303367, 3.85117540421915734118539905393, 4.32749048261804093995022526424, 4.44611033473386652078181611597, 4.52246026204136968592771709365, 4.76123984050685927758013431927, 4.99008084440527126610594509979, 5.02425844198564675425156190050, 5.37628810429703844266730570057, 5.63900081481304789680064521445, 5.64082083506837630840228247212, 5.90786669397225819413715186453, 6.43963969332001296391871019668, 6.59018359807203580378296515854

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