Properties

Label 8-1323e4-1.1-c3e4-0-0
Degree $8$
Conductor $3.064\times 10^{12}$
Sign $1$
Analytic cond. $3.71281\times 10^{7}$
Root an. cond. $8.83513$
Motivic weight $3$
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  − 3·4-s + 68·13-s + 25·16-s − 172·19-s − 114·25-s − 408·31-s + 1.32e3·37-s + 116·43-s − 204·52-s + 2.67e3·61-s − 315·64-s + 1.44e3·67-s + 396·73-s + 516·76-s + 1.22e3·79-s + 624·97-s + 342·100-s − 1.30e3·103-s + 720·109-s + 582·121-s + 1.22e3·124-s + 127-s + 131-s + 137-s + 139-s − 3.96e3·148-s + 149-s + ⋯
L(s)  = 1  − 3/8·4-s + 1.45·13-s + 0.390·16-s − 2.07·19-s − 0.911·25-s − 2.36·31-s + 5.86·37-s + 0.411·43-s − 0.544·52-s + 5.60·61-s − 0.615·64-s + 2.63·67-s + 0.634·73-s + 0.778·76-s + 1.73·79-s + 0.653·97-s + 0.341·100-s − 1.24·103-s + 0.632·109-s + 0.437·121-s + 0.886·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 2.19·148-s + 0.000549·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.71281\times 10^{7}\)
Root analytic conductor: \(8.83513\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{12} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(8.447076776\)
\(L(\frac12)\) \(\approx\) \(8.447076776\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$D_4\times C_2$ \( 1 + 3 T^{2} - p^{4} T^{4} + 3 p^{6} T^{6} + p^{12} T^{8} \)
5$D_4\times C_2$ \( 1 + 114 T^{2} + 25139 T^{4} + 114 p^{6} T^{6} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 582 T^{2} - 499957 T^{4} - 582 p^{6} T^{6} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 34 T - 582 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 9646 T^{2} + p^{6} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 86 T + 13227 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 334 p T^{2} + 253878819 T^{4} + 334 p^{7} T^{6} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 + 79152 T^{2} + 2674078958 T^{4} + 79152 p^{6} T^{6} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 + 204 T + 55361 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 660 T + 204941 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 4794 T^{2} + 6674554091 T^{4} + 4794 p^{6} T^{6} + p^{12} T^{8} \)
43$D_{4}$ \( ( 1 - 58 T + 159270 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 19688 T^{2} - 8519622546 T^{4} + 19688 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 26852 T^{2} + 7143822294 T^{4} + 26852 p^{6} T^{6} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 + 603752 T^{2} + 171721941918 T^{4} + 603752 p^{6} T^{6} + p^{12} T^{8} \)
61$D_{4}$ \( ( 1 - 1336 T + 890826 T^{2} - 1336 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 722 T + 520662 T^{2} - 722 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 582818 T^{2} + 161061490563 T^{4} + 582818 p^{6} T^{6} + p^{12} T^{8} \)
73$D_{4}$ \( ( 1 - 198 T + 773210 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 610 T + 980238 T^{2} - 610 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 1579208 T^{2} + 1195208061054 T^{4} + 1579208 p^{6} T^{6} + p^{12} T^{8} \)
89$D_4\times C_2$ \( 1 + 2521002 T^{2} + 2575576968923 T^{4} + 2521002 p^{6} T^{6} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 312 T + 1840322 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−6.56218254597625792967669205816, −6.36988191177389327727810923945, −5.91351902338233832174905264054, −5.77152848922439391695419823672, −5.67203426912053884853486149897, −5.58626466266196626954064722800, −5.18474163562930591268857888614, −4.83907158347323345339405052832, −4.74994257986462456538914680011, −4.30160803245176864651427985934, −4.07000984457363537420500899047, −4.03628000803736694765882855051, −3.81535765385454747994025872823, −3.57523787172336859884453511744, −3.47786240323766936653072639440, −2.79159577323772058150256617136, −2.58520971911799224825745820198, −2.47162418026906036769262843872, −2.06755728409460973295178542563, −1.97304883404062398980946370739, −1.58168507727422916655856884221, −0.941349331093177547140292894585, −0.917592372280660955496232910496, −0.49306999778494320234740779790, −0.46660114552583869593338181896, 0.46660114552583869593338181896, 0.49306999778494320234740779790, 0.917592372280660955496232910496, 0.941349331093177547140292894585, 1.58168507727422916655856884221, 1.97304883404062398980946370739, 2.06755728409460973295178542563, 2.47162418026906036769262843872, 2.58520971911799224825745820198, 2.79159577323772058150256617136, 3.47786240323766936653072639440, 3.57523787172336859884453511744, 3.81535765385454747994025872823, 4.03628000803736694765882855051, 4.07000984457363537420500899047, 4.30160803245176864651427985934, 4.74994257986462456538914680011, 4.83907158347323345339405052832, 5.18474163562930591268857888614, 5.58626466266196626954064722800, 5.67203426912053884853486149897, 5.77152848922439391695419823672, 5.91351902338233832174905264054, 6.36988191177389327727810923945, 6.56218254597625792967669205816

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