Properties

Label 8-1323e4-1.1-c1e4-0-2
Degree $8$
Conductor $3.064\times 10^{12}$
Sign $1$
Analytic cond. $12455.1$
Root an. cond. $3.25026$
Motivic weight $1$
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  − 2·4-s + 16·13-s − 3·16-s + 12·19-s − 14·25-s + 20·31-s − 4·37-s − 32·52-s + 8·61-s + 12·64-s + 40·73-s − 24·76-s − 16·79-s + 16·97-s + 28·100-s + 20·103-s − 12·109-s − 6·121-s − 40·124-s + 127-s + 131-s + 137-s + 139-s + 8·148-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 4-s + 4.43·13-s − 3/4·16-s + 2.75·19-s − 2.79·25-s + 3.59·31-s − 0.657·37-s − 4.43·52-s + 1.02·61-s + 3/2·64-s + 4.68·73-s − 2.75·76-s − 1.80·79-s + 1.62·97-s + 14/5·100-s + 1.97·103-s − 1.14·109-s − 0.545·121-s − 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(12455.1\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.816428656\)
\(L(\frac12)\) \(\approx\) \(4.816428656\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2 \wr C_2$ \( 1 + p T^{2} + 7 T^{4} + p^{3} T^{6} + p^{4} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 + 14 T^{2} + 97 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 6 T^{2} + 233 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 + 48 T^{2} + 1082 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 + 2 T^{2} + 1057 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 + 20 T^{2} + 1270 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 - 10 T + 55 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 2 T + 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 + 138 T^{2} + 7961 T^{4} + 138 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 36 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - 56 T^{2} + 5130 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 + 108 T^{2} + 5942 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 + 224 T^{2} + 19498 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - 10 T^{2} + 5305 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 - 20 T + 244 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 8 T + 76 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 + 116 T^{2} + 5590 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 + 42 T^{2} - 8359 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 8 T + 178 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.68630451777558793754329236746, −6.61710306976616621455701494829, −6.60246917122514352707098332572, −6.11252333019577611237154748251, −6.03261086903192966825208175934, −5.78878725162735550954208754453, −5.57005877073615517393790871555, −5.32211948054130676557195257473, −5.26374977711413510701342090043, −4.75967814185732876595708699491, −4.61966293064873693599803954730, −4.34228919780989842307551867763, −4.12855425516994439071979503820, −3.79403083013742057573715302133, −3.63160699092215927574436569516, −3.47935950519704869468560408985, −3.41103462561610339595769797872, −2.87584121943402312033224766960, −2.65513127308151700255219102590, −2.20475677507131938368984025300, −1.86063760019623894860488019417, −1.46860582479803359462865164825, −1.16384581348343585395474893801, −0.792466556965372655585308910725, −0.62841134942850118325255961102, 0.62841134942850118325255961102, 0.792466556965372655585308910725, 1.16384581348343585395474893801, 1.46860582479803359462865164825, 1.86063760019623894860488019417, 2.20475677507131938368984025300, 2.65513127308151700255219102590, 2.87584121943402312033224766960, 3.41103462561610339595769797872, 3.47935950519704869468560408985, 3.63160699092215927574436569516, 3.79403083013742057573715302133, 4.12855425516994439071979503820, 4.34228919780989842307551867763, 4.61966293064873693599803954730, 4.75967814185732876595708699491, 5.26374977711413510701342090043, 5.32211948054130676557195257473, 5.57005877073615517393790871555, 5.78878725162735550954208754453, 6.03261086903192966825208175934, 6.11252333019577611237154748251, 6.60246917122514352707098332572, 6.61710306976616621455701494829, 6.68630451777558793754329236746

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