Properties

Degree $8$
Conductor $7.966\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s − 6·7-s + 2·9-s + 8·15-s − 12·17-s + 12·21-s + 10·25-s − 6·27-s + 24·35-s + 12·37-s − 8·41-s − 16·43-s − 8·45-s − 8·47-s + 18·49-s + 24·51-s − 12·63-s + 48·67-s − 20·75-s + 11·81-s − 4·83-s + 48·85-s + 40·89-s + 32·101-s − 48·105-s − 24·111-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s − 2.26·7-s + 2/3·9-s + 2.06·15-s − 2.91·17-s + 2.61·21-s + 2·25-s − 1.15·27-s + 4.05·35-s + 1.97·37-s − 1.24·41-s − 2.43·43-s − 1.19·45-s − 1.16·47-s + 18/7·49-s + 3.36·51-s − 1.51·63-s + 5.86·67-s − 2.30·75-s + 11/9·81-s − 0.439·83-s + 5.20·85-s + 4.23·89-s + 3.18·101-s − 4.68·105-s − 2.27·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4628062428\)
\(L(\frac12)\) \(\approx\) \(0.4628062428\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_1$ \( ( 1 + T )^{4} \)
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$D_4\times C_2$ \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_4\times C_2$ \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
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$D_{4}$ \( ( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 + 2 T - 78 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.66498553078278138667165030181, −6.56106077076978569060464554775, −6.43840980167514747513930066702, −6.10654048442219512628034356132, −5.98279254890133807172330459390, −5.54702520194793154193023487394, −5.39246272762699435325855789867, −4.99654695194061838572790554390, −4.93131617687550189457514548608, −4.63762755889374355324945374256, −4.62978828902664395454577419583, −4.22393615570992350387789236651, −3.86167420635426188678849682444, −3.68074304791265306997459786622, −3.58697614332718477325193177654, −3.52823915824955971478033856130, −3.00524266907389298534696270949, −2.88987444622136810356448536851, −2.39748030781449297594046047643, −2.14114945594425393114103069121, −2.01706663257402181522387983089, −1.49463270926863549536525469824, −0.60884747366574297473818730934, −0.58678745570981596150205855784, −0.33422167430254205903067072716, 0.33422167430254205903067072716, 0.58678745570981596150205855784, 0.60884747366574297473818730934, 1.49463270926863549536525469824, 2.01706663257402181522387983089, 2.14114945594425393114103069121, 2.39748030781449297594046047643, 2.88987444622136810356448536851, 3.00524266907389298534696270949, 3.52823915824955971478033856130, 3.58697614332718477325193177654, 3.68074304791265306997459786622, 3.86167420635426188678849682444, 4.22393615570992350387789236651, 4.62978828902664395454577419583, 4.63762755889374355324945374256, 4.93131617687550189457514548608, 4.99654695194061838572790554390, 5.39246272762699435325855789867, 5.54702520194793154193023487394, 5.98279254890133807172330459390, 6.10654048442219512628034356132, 6.43840980167514747513930066702, 6.56106077076978569060464554775, 6.66498553078278138667165030181

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