Properties

Label 8-1344e4-1.1-c2e4-0-4
Degree $8$
Conductor $3.263\times 10^{12}$
Sign $1$
Analytic cond. $1.79861\times 10^{6}$
Root an. cond. $6.05155$
Motivic weight $2$
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  − 8·7-s − 6·9-s − 24·11-s − 24·23-s + 64·25-s − 120·29-s + 80·37-s − 128·43-s + 22·49-s + 216·53-s + 48·63-s + 176·67-s + 120·71-s + 192·77-s − 128·79-s + 27·81-s + 144·99-s − 168·107-s + 8·109-s − 360·113-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 8/7·7-s − 2/3·9-s − 2.18·11-s − 1.04·23-s + 2.55·25-s − 4.13·29-s + 2.16·37-s − 2.97·43-s + 0.448·49-s + 4.07·53-s + 0.761·63-s + 2.62·67-s + 1.69·71-s + 2.49·77-s − 1.62·79-s + 1/3·81-s + 1.45·99-s − 1.57·107-s + 0.0733·109-s − 3.18·113-s − 0.727·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.79861\times 10^{6}\)
Root analytic conductor: \(6.05155\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5225845151\)
\(L(\frac12)\) \(\approx\) \(0.5225845151\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 8 T + 6 p T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \)
good5$D_4\times C_2$ \( 1 - 64 T^{2} + 1986 T^{4} - 64 p^{4} T^{6} + p^{8} T^{8} \)
11$D_{4}$ \( ( 1 + 12 T + 260 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 244 T^{2} + 53574 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} \)
17$C_2^3$ \( 1 + 320 T^{2} + 546 p^{2} T^{4} + 320 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 652 T^{2} + 348486 T^{4} - 652 p^{4} T^{6} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 + 12 T + 932 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{4} \)
31$D_4\times C_2$ \( 1 - 1252 T^{2} + 745926 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 40 T + 546 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 4960 T^{2} + 11110434 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 64 T + 2922 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 3940 T^{2} + 5766 p^{2} T^{4} - 3940 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 108 T + 8246 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 4420 T^{2} + 6539622 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 14596 T^{2} + 80934054 T^{4} - 14596 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 88 T + 10626 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 60 T + 4484 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 13108 T^{2} + 98972646 T^{4} - 13108 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 64 T + 3138 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 16612 T^{2} + 134415078 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 8896 T^{2} + 52680354 T^{4} - 8896 p^{4} T^{6} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 24820 T^{2} + 317127462 T^{4} - 24820 p^{4} T^{6} + p^{8} T^{8} \)
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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

−6.66121100686072521521821407269, −6.41865182798310497144751210601, −6.19182608296560685135352647052, −6.13905730171918227357883531475, −5.58663539995694140345922197731, −5.43373342390183975313227739943, −5.37651490066744362646048443407, −5.30118625895994565522213005321, −4.97704024230307590584101326808, −4.87276500426580781272480595782, −4.31797784258718492498431686668, −3.94548569413904397002695092774, −3.80010455219278048698967493114, −3.78930359592666753589596928377, −3.54924719395106220760448034082, −2.95891904323978179449871289618, −2.76460589225667875484154077062, −2.71811408849683900423507274506, −2.38254278604623557866628344324, −2.22072192854790041848568479777, −1.76772055867422893577125506541, −1.35938692557850369150529957440, −0.925706929010576570750848355139, −0.41597601359052552697920515154, −0.17160348618925721664823209766, 0.17160348618925721664823209766, 0.41597601359052552697920515154, 0.925706929010576570750848355139, 1.35938692557850369150529957440, 1.76772055867422893577125506541, 2.22072192854790041848568479777, 2.38254278604623557866628344324, 2.71811408849683900423507274506, 2.76460589225667875484154077062, 2.95891904323978179449871289618, 3.54924719395106220760448034082, 3.78930359592666753589596928377, 3.80010455219278048698967493114, 3.94548569413904397002695092774, 4.31797784258718492498431686668, 4.87276500426580781272480595782, 4.97704024230307590584101326808, 5.30118625895994565522213005321, 5.37651490066744362646048443407, 5.43373342390183975313227739943, 5.58663539995694140345922197731, 6.13905730171918227357883531475, 6.19182608296560685135352647052, 6.41865182798310497144751210601, 6.66121100686072521521821407269

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