Properties

Label 8-1560e4-1.1-c0e4-0-3
Degree $8$
Conductor $5.922\times 10^{12}$
Sign $1$
Analytic cond. $0.367389$
Root an. cond. $0.882349$
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  + 4·3-s + 10·9-s − 16-s − 4·17-s + 20·27-s + 4·43-s − 4·48-s − 16·51-s + 35·81-s − 4·107-s + 4·113-s − 4·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s − 40·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4·3-s + 10·9-s − 16-s − 4·17-s + 20·27-s + 4·43-s − 4·48-s − 16·51-s + 35·81-s − 4·107-s + 4·113-s − 4·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s − 40·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.367389\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.539317858\)
\(L(\frac12)\) \(\approx\) \(5.539317858\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−7.01842154701136052638403789465, −6.83189348450268094654289111670, −6.82638918579746069196331755961, −6.49806366025205173377229351045, −6.06013572796187894318966208693, −5.97859421161292538745308059314, −5.95858762495327392133573555824, −4.98146631992474167106096851895, −4.94805255767929023489199281854, −4.91908464627980076951753285388, −4.54211115151797703555642602882, −4.33158477315213165976867180788, −4.03957937575070272124451128453, −4.03450169591669412052882930906, −3.87653230167456957326016644601, −3.57578624758234445644758624542, −3.12210891156062601440413911045, −2.96753746262196121044430277518, −2.43682870814301258940203836563, −2.40121690528520517899430977326, −2.39858619384828046134713780174, −2.26622689572498962660598281407, −1.81470004441514131644055188475, −1.25763938007383682012569400836, −1.19233075462391775007203047007, 1.19233075462391775007203047007, 1.25763938007383682012569400836, 1.81470004441514131644055188475, 2.26622689572498962660598281407, 2.39858619384828046134713780174, 2.40121690528520517899430977326, 2.43682870814301258940203836563, 2.96753746262196121044430277518, 3.12210891156062601440413911045, 3.57578624758234445644758624542, 3.87653230167456957326016644601, 4.03450169591669412052882930906, 4.03957937575070272124451128453, 4.33158477315213165976867180788, 4.54211115151797703555642602882, 4.91908464627980076951753285388, 4.94805255767929023489199281854, 4.98146631992474167106096851895, 5.95858762495327392133573555824, 5.97859421161292538745308059314, 6.06013572796187894318966208693, 6.49806366025205173377229351045, 6.82638918579746069196331755961, 6.83189348450268094654289111670, 7.01842154701136052638403789465

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