Properties

Degree $8$
Conductor $1.000\times 10^{12}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 4·9-s − 2·11-s + 3·16-s + 2·19-s + 8·36-s − 2·41-s + 4·44-s + 49-s + 2·59-s − 4·64-s − 4·76-s + 10·81-s + 2·89-s + 8·99-s + 121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + 167-s + ⋯
L(s)  = 1  − 2·4-s − 4·9-s − 2·11-s + 3·16-s + 2·19-s + 8·36-s − 2·41-s + 4·44-s + 49-s + 2·59-s − 4·64-s − 4·76-s + 10·81-s + 2·89-s + 8·99-s + 121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{12}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{12} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2074012741\)
\(L(\frac12)\) \(\approx\) \(0.2074012741\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
good3$C_2$ \( ( 1 + T^{2} )^{4} \)
7$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
11$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
53$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
59$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−7.52763351210575740728574484470, −7.27168070509826751643936382108, −6.96030708153170355150733223390, −6.64651576831471675882014119021, −6.27597403158564102147500296468, −6.03309163557421023811755445472, −5.95425057796081598403791478785, −5.52315502847623226942703368889, −5.51017068280432819385862507952, −5.27803604498195023920795062366, −5.11089164664135601252674248407, −5.08697837931114319036111818488, −4.82706710304148869006178771927, −4.35950311828659390244930121243, −4.05044386761420648592790285583, −3.57312699061533403071699906158, −3.55732012453145747468403924025, −3.23438279179993723560700966856, −3.04839952210451431850416556123, −2.80119684492478915363972911188, −2.50502165784129229413176958115, −2.24649263784152407964293391754, −1.69926707956014038299708096864, −0.886336849121244270002008400291, −0.48110794230330819991389489200, 0.48110794230330819991389489200, 0.886336849121244270002008400291, 1.69926707956014038299708096864, 2.24649263784152407964293391754, 2.50502165784129229413176958115, 2.80119684492478915363972911188, 3.04839952210451431850416556123, 3.23438279179993723560700966856, 3.55732012453145747468403924025, 3.57312699061533403071699906158, 4.05044386761420648592790285583, 4.35950311828659390244930121243, 4.82706710304148869006178771927, 5.08697837931114319036111818488, 5.11089164664135601252674248407, 5.27803604498195023920795062366, 5.51017068280432819385862507952, 5.52315502847623226942703368889, 5.95425057796081598403791478785, 6.03309163557421023811755445472, 6.27597403158564102147500296468, 6.64651576831471675882014119021, 6.96030708153170355150733223390, 7.27168070509826751643936382108, 7.52763351210575740728574484470

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