Properties

Label 8-325e4-1.1-c1e4-0-5
Degree $8$
Conductor $11156640625$
Sign $1$
Analytic cond. $45.3567$
Root an. cond. $1.61094$
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  + 2·4-s + 8·9-s + 8·11-s + 3·16-s − 8·19-s + 24·31-s + 16·36-s − 24·41-s + 16·44-s + 4·49-s − 24·59-s − 32·61-s + 12·64-s + 8·71-s − 16·76-s + 30·81-s − 24·89-s + 64·99-s + 8·101-s + 8·109-s + 48·124-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + ⋯
L(s)  = 1  + 4-s + 8/3·9-s + 2.41·11-s + 3/4·16-s − 1.83·19-s + 4.31·31-s + 8/3·36-s − 3.74·41-s + 2.41·44-s + 4/7·49-s − 3.12·59-s − 4.09·61-s + 3/2·64-s + 0.949·71-s − 1.83·76-s + 10/3·81-s − 2.54·89-s + 6.43·99-s + 0.796·101-s + 0.766·109-s + 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(45.3567\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.263543212\)
\(L(\frac12)\) \(\approx\) \(4.263543212\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
3$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
7$C_4\times C_2$ \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 40 T^{2} + 898 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 164 T^{2} + 11014 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 4 T^{2} - 4746 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 244 T^{2} + 27510 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \)
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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

−8.415129202032320510573603104190, −8.326247785816936881571428467600, −7.932178960998732661205011392578, −7.56380664135905125363901841273, −7.43290435465766031892666665230, −6.86418811235366892923923043799, −6.84947671251294459386375328605, −6.74166187094870676510474222068, −6.41048873193910500344615079808, −6.38670858446185310427143492566, −5.99639806895456131384643829312, −5.81103168564962418863585531299, −5.05969485459207818188834008759, −4.80720084754835869289367418537, −4.52194186991981177936123514934, −4.40416558346710787350732670691, −4.13391505120304211576385687847, −3.88857481869359943309003123182, −3.35199497205617242470145596643, −2.99805147309496647866015010285, −2.81902202850165259968279460448, −1.87845701380858593970588463932, −1.78844196603554217417148630014, −1.48209876826564265259377156858, −1.03269878870999963998802987334, 1.03269878870999963998802987334, 1.48209876826564265259377156858, 1.78844196603554217417148630014, 1.87845701380858593970588463932, 2.81902202850165259968279460448, 2.99805147309496647866015010285, 3.35199497205617242470145596643, 3.88857481869359943309003123182, 4.13391505120304211576385687847, 4.40416558346710787350732670691, 4.52194186991981177936123514934, 4.80720084754835869289367418537, 5.05969485459207818188834008759, 5.81103168564962418863585531299, 5.99639806895456131384643829312, 6.38670858446185310427143492566, 6.41048873193910500344615079808, 6.74166187094870676510474222068, 6.84947671251294459386375328605, 6.86418811235366892923923043799, 7.43290435465766031892666665230, 7.56380664135905125363901841273, 7.932178960998732661205011392578, 8.326247785816936881571428467600, 8.415129202032320510573603104190

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