Properties

Label 8-825e4-1.1-c1e4-0-8
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s + 8·7-s − 12·8-s + 2·9-s + 8·13-s − 32·14-s + 15·16-s − 8·18-s − 4·23-s − 32·26-s + 64·28-s + 8·29-s − 24·31-s − 16·32-s + 16·36-s + 12·37-s + 16·43-s + 16·46-s − 12·47-s + 32·49-s + 64·52-s + 4·53-s − 96·56-s − 32·58-s + 16·59-s + 8·61-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s + 3.02·7-s − 4.24·8-s + 2/3·9-s + 2.21·13-s − 8.55·14-s + 15/4·16-s − 1.88·18-s − 0.834·23-s − 6.27·26-s + 12.0·28-s + 1.48·29-s − 4.31·31-s − 2.82·32-s + 8/3·36-s + 1.97·37-s + 2.43·43-s + 2.35·46-s − 1.75·47-s + 32/7·49-s + 8.87·52-s + 0.549·53-s − 12.8·56-s − 4.20·58-s + 2.08·59-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.256183167\)
\(L(\frac12)\) \(\approx\) \(1.256183167\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5 \( 1 \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{3} T^{5} + p^{5} T^{6} + p^{5} T^{7} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 254 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} - 44 T^{3} - 914 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 468 T^{3} + 3038 T^{4} - 468 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 92 T^{2} + 4966 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 16 T + 128 T^{2} - 624 T^{3} + 3026 T^{4} - 624 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 348 T^{3} + 1358 T^{4} + 348 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 2260 T^{3} + 23422 T^{4} - 2260 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 68 T^{2} + 870 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 172 T^{2} + 17830 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} - 72 T^{3} - 8302 T^{4} - 72 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 13582 T^{4} + 1660 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.39515131259821647017444945482, −7.18585303658935740544432848513, −7.13500561494779272224453310191, −7.06790303383958444493729028924, −6.44633939726195695453190438755, −6.23017491468685556575326528040, −6.16731434413030496508967783157, −5.77662101769641470543053756919, −5.36942312791163025134934762212, −5.32669152771374113443053634103, −5.27950012327672961362197637025, −4.68714575699672639251431252438, −4.40573890163771643283712766748, −4.22074830615998325008382276256, −3.80131330969066967900311546767, −3.75751367762212821207982499877, −3.52040136551838228384131183023, −2.79307672292130502550070732108, −2.43294044064145393066262610440, −2.23183340527645237481113516078, −1.75735816350239481435046658463, −1.63304861552945845191029180550, −1.34696211033206194750660556907, −0.842624576763424979892029805874, −0.62600548018974399612658021614, 0.62600548018974399612658021614, 0.842624576763424979892029805874, 1.34696211033206194750660556907, 1.63304861552945845191029180550, 1.75735816350239481435046658463, 2.23183340527645237481113516078, 2.43294044064145393066262610440, 2.79307672292130502550070732108, 3.52040136551838228384131183023, 3.75751367762212821207982499877, 3.80131330969066967900311546767, 4.22074830615998325008382276256, 4.40573890163771643283712766748, 4.68714575699672639251431252438, 5.27950012327672961362197637025, 5.32669152771374113443053634103, 5.36942312791163025134934762212, 5.77662101769641470543053756919, 6.16731434413030496508967783157, 6.23017491468685556575326528040, 6.44633939726195695453190438755, 7.06790303383958444493729028924, 7.13500561494779272224453310191, 7.18585303658935740544432848513, 7.39515131259821647017444945482

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