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Properties

Label 8-425e4-1.1-c1e4-0-1
Degree $8$
Conductor $32625390625$
Sign $1$
Analytic cond. $132.636$
Root an. cond. $1.84218$
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  + 2·2-s + 4·3-s + 8·6-s + 10·7-s − 4·8-s + 4·9-s − 2·11-s + 6·13-s + 20·14-s − 7·16-s − 4·17-s + 8·18-s + 4·19-s + 40·21-s − 4·22-s + 4·23-s − 16·24-s + 12·26-s − 6·27-s − 4·29-s − 12·31-s − 4·32-s − 8·33-s − 8·34-s + 12·37-s + 8·38-s + 24·39-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 3.26·6-s + 3.77·7-s − 1.41·8-s + 4/3·9-s − 0.603·11-s + 1.66·13-s + 5.34·14-s − 7/4·16-s − 0.970·17-s + 1.88·18-s + 0.917·19-s + 8.72·21-s − 0.852·22-s + 0.834·23-s − 3.26·24-s + 2.35·26-s − 1.15·27-s − 0.742·29-s − 2.15·31-s − 0.707·32-s − 1.39·33-s − 1.37·34-s + 1.97·37-s + 1.29·38-s + 3.84·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(132.636\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.94626351\)
\(L(\frac12)\) \(\approx\) \(13.94626351\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_1$ \( ( 1 + T )^{4} \)
good2$C_2 \wr S_4$ \( 1 - p T + p^{2} T^{2} - p^{2} T^{3} + 7 T^{4} - p^{3} T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
3$C_2 \wr S_4$ \( 1 - 4 T + 4 p T^{2} - 26 T^{3} + 52 T^{4} - 26 p T^{5} + 4 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 10 T + 60 T^{2} - 248 T^{3} + 108 p T^{4} - 248 p T^{5} + 60 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 2 T + 30 T^{2} + 40 T^{3} + 420 T^{4} + 40 p T^{5} + 30 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 6 T + 24 T^{2} - 42 T^{3} + 122 T^{4} - 42 p T^{5} + 24 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 4 T + 44 T^{2} - 148 T^{3} + 1062 T^{4} - 148 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 4 T + 72 T^{2} - 194 T^{3} + 2244 T^{4} - 194 p T^{5} + 72 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 4 T + 84 T^{2} + 204 T^{3} + 3110 T^{4} + 204 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 12 T + 118 T^{2} + 926 T^{3} + 5468 T^{4} + 926 p T^{5} + 118 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 12 T + 164 T^{2} - 1284 T^{3} + 9382 T^{4} - 1284 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 6 T + 128 T^{2} + 586 T^{3} + 7526 T^{4} + 586 p T^{5} + 128 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 18 T + 268 T^{2} - 2410 T^{3} + 19034 T^{4} - 2410 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 6 T + 160 T^{2} - 654 T^{3} + 10458 T^{4} - 654 p T^{5} + 160 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 8 T + 108 T^{2} - 536 T^{3} + 5846 T^{4} - 536 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
61$C_2 \wr S_4$ \( 1 - 6 T + 152 T^{2} - 762 T^{3} + 12982 T^{4} - 762 p T^{5} + 152 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 6 T + 216 T^{2} - 990 T^{3} + 19914 T^{4} - 990 p T^{5} + 216 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 10 T + 266 T^{2} + 1932 T^{3} + 27932 T^{4} + 1932 p T^{5} + 266 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 2 T + 116 T^{2} - 262 T^{3} + 7534 T^{4} - 262 p T^{5} + 116 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 12 T + 258 T^{2} + 2274 T^{3} + 27756 T^{4} + 2274 p T^{5} + 258 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 14 T + 288 T^{2} + 2582 T^{3} + 33074 T^{4} + 2582 p T^{5} + 288 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 24 T + 492 T^{2} - 6276 T^{3} + 71250 T^{4} - 6276 p T^{5} + 492 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 4 T + 268 T^{2} + 12 p T^{3} + 35174 T^{4} + 12 p^{2} T^{5} + 268 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.407686241715306813218527232053, −7.70373378113374966797742392502, −7.59506586241730883142587697928, −7.57516510337933116993685058450, −7.51617438955552349995108202393, −7.06861791761670270864053051395, −6.48130414230619708124518394423, −6.10716470395621445229405114258, −6.07304683077331756232203957722, −5.49590413185336397921764394912, −5.39216983831768340094529020672, −5.31025430103094668782738737934, −4.90332292582932388158167201044, −4.65949291487394890542828451166, −4.31794555354980880037923099556, −4.10887204554274865993143797860, −4.09945296375254821657498890906, −3.49934010201651181428836868019, −3.20836939877178302429076573018, −3.05995027922568997721690164060, −2.44492992089778341553246388337, −2.26438068374847289591696769148, −2.02271564153066031163322190915, −1.49374934876020897984493821793, −1.00808006538301713745123418662, 1.00808006538301713745123418662, 1.49374934876020897984493821793, 2.02271564153066031163322190915, 2.26438068374847289591696769148, 2.44492992089778341553246388337, 3.05995027922568997721690164060, 3.20836939877178302429076573018, 3.49934010201651181428836868019, 4.09945296375254821657498890906, 4.10887204554274865993143797860, 4.31794555354980880037923099556, 4.65949291487394890542828451166, 4.90332292582932388158167201044, 5.31025430103094668782738737934, 5.39216983831768340094529020672, 5.49590413185336397921764394912, 6.07304683077331756232203957722, 6.10716470395621445229405114258, 6.48130414230619708124518394423, 7.06861791761670270864053051395, 7.51617438955552349995108202393, 7.57516510337933116993685058450, 7.59506586241730883142587697928, 7.70373378113374966797742392502, 8.407686241715306813218527232053

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