Properties

Degree $8$
Conductor $1.216\times 10^{12}$
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  − 4·2-s − 2·3-s + 10·4-s + 8·6-s + 6·7-s − 20·8-s + 2·9-s − 20·12-s − 4·13-s − 24·14-s + 35·16-s − 8·18-s − 12·21-s − 16·23-s + 40·24-s + 16·26-s − 6·27-s + 60·28-s − 56·32-s + 20·36-s + 8·39-s − 8·41-s + 48·42-s + 64·46-s − 70·48-s + 18·49-s − 40·52-s + ⋯
L(s)  = 1  − 2.82·2-s − 1.15·3-s + 5·4-s + 3.26·6-s + 2.26·7-s − 7.07·8-s + 2/3·9-s − 5.77·12-s − 1.10·13-s − 6.41·14-s + 35/4·16-s − 1.88·18-s − 2.61·21-s − 3.33·23-s + 8.16·24-s + 3.13·26-s − 1.15·27-s + 11.3·28-s − 9.89·32-s + 10/3·36-s + 1.28·39-s − 1.24·41-s + 7.40·42-s + 9.43·46-s − 10.1·48-s + 18/7·49-s − 5.54·52-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2961959954\)
\(L(\frac12)\) \(\approx\) \(0.2961959954\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 40 T^{2} + 798 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
19$C_4\times C_2$ \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
29$D_4\times C_2$ \( 1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 120 T^{2} + 6158 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 20 T^{2} - 1322 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 140 T^{2} + 8998 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 184 T^{2} + 15406 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 160 T^{2} + 19198 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 6 T + 198 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.20412132874865593758765174077, −7.06436220287970016330798350741, −6.82670224911881057051426153471, −6.43890368094671634597026462240, −6.25438630348587563263632508419, −6.04607466821190160550480782195, −5.99613106683636009633390000974, −5.57456832369174855940951363830, −5.42394924641096959119096631266, −5.03474360345807052703714528830, −5.00078550630960293214710899284, −4.62375780971466794182333738391, −4.28178024972992515984636792093, −4.26671197495530365215800840917, −3.58333439785814228139229620127, −3.42369079984984348551285982004, −3.31107826130635190812698025934, −2.36658827240389304519261698806, −2.29487039322055280608052430555, −2.28802839468991303268469151339, −1.78535800500740185151907143042, −1.49295156565076076080166328651, −1.42681952309950710163665546073, −0.57567379642023884554996788649, −0.34090529985331480129948075856, 0.34090529985331480129948075856, 0.57567379642023884554996788649, 1.42681952309950710163665546073, 1.49295156565076076080166328651, 1.78535800500740185151907143042, 2.28802839468991303268469151339, 2.29487039322055280608052430555, 2.36658827240389304519261698806, 3.31107826130635190812698025934, 3.42369079984984348551285982004, 3.58333439785814228139229620127, 4.26671197495530365215800840917, 4.28178024972992515984636792093, 4.62375780971466794182333738391, 5.00078550630960293214710899284, 5.03474360345807052703714528830, 5.42394924641096959119096631266, 5.57456832369174855940951363830, 5.99613106683636009633390000974, 6.04607466821190160550480782195, 6.25438630348587563263632508419, 6.43890368094671634597026462240, 6.82670224911881057051426153471, 7.06436220287970016330798350741, 7.20412132874865593758765174077

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