Properties

Label 8-1216e4-1.1-c0e4-0-2
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  − 2·5-s − 9-s + 2·13-s − 2·17-s + 3·25-s + 2·29-s − 2·41-s + 2·45-s + 4·49-s − 2·53-s − 2·61-s − 4·65-s + 2·73-s + 81-s + 4·85-s − 2·89-s − 2·97-s + 2·101-s + 2·109-s − 2·117-s + 4·121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯
L(s)  = 1  − 2·5-s − 9-s + 2·13-s − 2·17-s + 3·25-s + 2·29-s − 2·41-s + 2·45-s + 4·49-s − 2·53-s − 2·61-s − 4·65-s + 2·73-s + 81-s + 4·85-s − 2·89-s − 2·97-s + 2·101-s + 2·109-s − 2·117-s + 4·121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-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

\(L(\frac{1}{2})\) \(\approx\) \(0.5323935134\)
\(L(\frac12)\) \(\approx\) \(0.5323935134\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
59$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
79$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
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

−7.29342028013455314973685785391, −6.75182038799543721294901214089, −6.72814245635822106468757900617, −6.72552270229868958113104509537, −6.18390062174837916117022491537, −6.18204788887165868167193491315, −6.00095206692150054700176195975, −5.66622558563739064601229591529, −5.30924291287216351748220960892, −5.10737313499839080746870562739, −4.71071284471956799887229606276, −4.68661615658372845467334466105, −4.51414379946950718010219316691, −4.05246587633755526302103191773, −3.98271308439624285514057492539, −3.74725239184903231767558580199, −3.39141442100870189581158695675, −3.21647588991003675215054211196, −3.01058750954397814600227403540, −2.57476138032715204162864717431, −2.46690302912986844679491042576, −1.97461444169619673993998852549, −1.50863776185989750114429940837, −1.10050280817860589540909830881, −0.59137151464952032825805018407, 0.59137151464952032825805018407, 1.10050280817860589540909830881, 1.50863776185989750114429940837, 1.97461444169619673993998852549, 2.46690302912986844679491042576, 2.57476138032715204162864717431, 3.01058750954397814600227403540, 3.21647588991003675215054211196, 3.39141442100870189581158695675, 3.74725239184903231767558580199, 3.98271308439624285514057492539, 4.05246587633755526302103191773, 4.51414379946950718010219316691, 4.68661615658372845467334466105, 4.71071284471956799887229606276, 5.10737313499839080746870562739, 5.30924291287216351748220960892, 5.66622558563739064601229591529, 6.00095206692150054700176195975, 6.18204788887165868167193491315, 6.18390062174837916117022491537, 6.72552270229868958113104509537, 6.72814245635822106468757900617, 6.75182038799543721294901214089, 7.29342028013455314973685785391

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