Properties

Label 8-882e4-1.1-c2e4-0-15
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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  + 4·4-s + 36·11-s + 12·16-s + 60·23-s + 46·25-s − 48·29-s + 124·37-s − 8·43-s + 144·44-s − 156·53-s + 32·64-s + 116·67-s + 24·71-s − 220·79-s + 240·92-s + 184·100-s + 132·107-s − 140·109-s − 240·113-s − 192·116-s + 362·121-s + 127-s + 131-s + 137-s + 139-s + 496·148-s + 149-s + ⋯
L(s)  = 1  + 4-s + 3.27·11-s + 3/4·16-s + 2.60·23-s + 1.83·25-s − 1.65·29-s + 3.35·37-s − 0.186·43-s + 3.27·44-s − 2.94·53-s + 1/2·64-s + 1.73·67-s + 0.338·71-s − 2.78·79-s + 2.60·92-s + 1.83·100-s + 1.23·107-s − 1.28·109-s − 2.12·113-s − 1.65·116-s + 2.99·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.35·148-s + 0.00671·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(10.99876730\)
\(L(\frac12)\) \(\approx\) \(10.99876730\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} ) \)
11$D_{4}$ \( ( 1 - 18 T + 305 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 412 T^{2} + 89190 T^{4} - 412 p^{4} T^{6} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 958 T^{2} + 1347 p^{2} T^{4} - 958 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 1426 T^{2} + 768939 T^{4} - 1426 p^{4} T^{6} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 - 30 T + 1121 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 24 T + 1754 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 850 T^{2} + 1233867 T^{4} - 850 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 62 T + 2547 T^{2} - 62 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 5500 T^{2} + 13185222 T^{4} - 5500 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 4 T + 3630 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 3778 T^{2} + 13267131 T^{4} - 3778 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 78 T + 6851 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 5410 T^{2} + 23946747 T^{4} - 5410 p^{4} T^{6} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 2302 T^{2} + 25403811 T^{4} - 2302 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 58 T + 5769 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 12 T + 8318 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 1390 T^{2} + 5504019 T^{4} - 1390 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 110 T + 15057 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 380 T^{2} + 89625894 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 958 T^{2} - 38888445 T^{4} - 958 p^{4} T^{6} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 26620 T^{2} + 330657414 T^{4} - 26620 p^{4} T^{6} + p^{8} T^{8} \)
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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

−7.08857239898925283578000204292, −6.76726558776477758564886374600, −6.48380343878567792125329679346, −6.46901925759053958925735196214, −6.33723419292844737019515519918, −6.09732967932960334635860894503, −5.79619945109085466932967662210, −5.32585581576011778826923744071, −5.27504741678829683268558853354, −5.02431952444926802774227512701, −4.56309047817362692800040330197, −4.45323018494937787248372736260, −4.26808845940740115553345564710, −3.67986816245922128510924033765, −3.64837622238910515353997446806, −3.64265849885300303281753492053, −2.92739685115207037772064566134, −2.73686412209988003438707155128, −2.73148707238963296827553810428, −2.27697421837978023070347830913, −1.50288773400785590953261439620, −1.44320860688683736308286188994, −1.34265330843990702305445438404, −0.952834671346080340450906598792, −0.45597250173844163118861628923, 0.45597250173844163118861628923, 0.952834671346080340450906598792, 1.34265330843990702305445438404, 1.44320860688683736308286188994, 1.50288773400785590953261439620, 2.27697421837978023070347830913, 2.73148707238963296827553810428, 2.73686412209988003438707155128, 2.92739685115207037772064566134, 3.64265849885300303281753492053, 3.64837622238910515353997446806, 3.67986816245922128510924033765, 4.26808845940740115553345564710, 4.45323018494937787248372736260, 4.56309047817362692800040330197, 5.02431952444926802774227512701, 5.27504741678829683268558853354, 5.32585581576011778826923744071, 5.79619945109085466932967662210, 6.09732967932960334635860894503, 6.33723419292844737019515519918, 6.46901925759053958925735196214, 6.48380343878567792125329679346, 6.76726558776477758564886374600, 7.08857239898925283578000204292

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