Properties

Label 8-1216e4-1.1-c0e4-0-1
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $0.135632$
Root an. cond. $0.779014$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 4·17-s − 2·25-s + 6·41-s − 4·49-s − 2·73-s + 81-s + 6·97-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 9-s − 4·17-s − 2·25-s + 6·41-s − 4·49-s − 2·73-s + 81-s + 6·97-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(0.135632\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5113773246\)
\(L(\frac12)\) \(\approx\) \(0.5113773246\)
\(L(1)\) 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 \( 1 \)
19$C_2^2$ \( 1 - T^{2} + T^{4} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2$ \( ( 1 + T + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \)
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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

−7.15159434707151250042789265430, −6.97711602030787978597865787828, −6.64648105408904582220075486021, −6.33257708674911452047802337702, −6.24973198387071939405427140802, −6.10045168952924732326500265181, −6.07511311539990027229873017074, −5.63838339759436784657694854294, −5.53776359687503644312219280328, −5.08077694003217113156156636332, −4.80308151849610231064714743141, −4.63205338091576310812121474394, −4.34459692288072221695634179016, −4.27618592708093664970233098100, −4.14816371993926317542914870285, −3.77572869709346389763175668568, −3.20312060153324186331548047291, −3.14486198115345709746484606932, −3.00488165892666332850729556221, −2.33145767731444444691075459073, −2.21709606127198372665030073163, −2.09417791663724743545999534168, −1.92954769839779621893925985129, −1.16938147683957324175620717374, −0.49780959635767278189143083260, 0.49780959635767278189143083260, 1.16938147683957324175620717374, 1.92954769839779621893925985129, 2.09417791663724743545999534168, 2.21709606127198372665030073163, 2.33145767731444444691075459073, 3.00488165892666332850729556221, 3.14486198115345709746484606932, 3.20312060153324186331548047291, 3.77572869709346389763175668568, 4.14816371993926317542914870285, 4.27618592708093664970233098100, 4.34459692288072221695634179016, 4.63205338091576310812121474394, 4.80308151849610231064714743141, 5.08077694003217113156156636332, 5.53776359687503644312219280328, 5.63838339759436784657694854294, 6.07511311539990027229873017074, 6.10045168952924732326500265181, 6.24973198387071939405427140802, 6.33257708674911452047802337702, 6.64648105408904582220075486021, 6.97711602030787978597865787828, 7.15159434707151250042789265430

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