Properties

Label 8-825e4-1.1-c1e4-0-12
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 − 4·3-s + 8·4-s + 16·6-s + 8·7-s − 12·8-s + 8·9-s − 32·12-s + 8·13-s − 32·14-s + 15·16-s + 8·17-s − 32·18-s − 32·21-s − 8·23-s + 48·24-s − 32·26-s − 12·27-s + 64·28-s + 24·29-s + 16·31-s − 16·32-s − 32·34-s + 64·36-s − 32·39-s + 128·42-s + 24·43-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 4·4-s + 6.53·6-s + 3.02·7-s − 4.24·8-s + 8/3·9-s − 9.23·12-s + 2.21·13-s − 8.55·14-s + 15/4·16-s + 1.94·17-s − 7.54·18-s − 6.98·21-s − 1.66·23-s + 9.79·24-s − 6.27·26-s − 2.30·27-s + 12.0·28-s + 4.45·29-s + 2.87·31-s − 2.82·32-s − 5.48·34-s + 32/3·36-s − 5.12·39-s + 19.7·42-s + 3.65·43-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

\(L(1)\) \(\approx\) \(0.7511474678\)
\(L(\frac12)\) \(\approx\) \(0.7511474678\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 136 T^{3} + 562 T^{4} - 136 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 386 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2638 T^{4} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 24 T + 288 T^{2} - 2664 T^{3} + 20018 T^{4} - 2664 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 312 T^{3} + 2978 T^{4} - 312 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 4174 T^{4} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 1520 T^{3} + 17266 T^{4} + 1520 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 188 T^{2} + 16870 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 24 T + 288 T^{2} - 3384 T^{3} + 35138 T^{4} - 3384 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^3$ \( 1 + 3122 T^{4} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 16 T + 128 T^{2} - 1040 T^{3} + 7426 T^{4} - 1040 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.80314647125066786448993389807, −6.98993014996485522428704601769, −6.98818305086864026162047085561, −6.74941015899702364386363172714, −6.46767298542277838367340735636, −6.07087848203495058771220942977, −6.00156105197476871734429072603, −5.92384062446878245856499389964, −5.62059734705143374752639771383, −5.47438072383007825770810311955, −4.95899549815059274624031668597, −4.76213534966201578063245969677, −4.63228425581758898541346514170, −4.38176797274964553257110051532, −4.13623703373338012358612309492, −3.67640512031733667831323361937, −3.44826203291104092713170521539, −2.68856413628720595144683254850, −2.61991069589528242279021791052, −2.19937499842792845011248697576, −1.75872019986227953391890441408, −1.13940940090919051705858296719, −1.04395488729028226688049480373, −0.911468997195144285650997953800, −0.74620254726863351706028655764, 0.74620254726863351706028655764, 0.911468997195144285650997953800, 1.04395488729028226688049480373, 1.13940940090919051705858296719, 1.75872019986227953391890441408, 2.19937499842792845011248697576, 2.61991069589528242279021791052, 2.68856413628720595144683254850, 3.44826203291104092713170521539, 3.67640512031733667831323361937, 4.13623703373338012358612309492, 4.38176797274964553257110051532, 4.63228425581758898541346514170, 4.76213534966201578063245969677, 4.95899549815059274624031668597, 5.47438072383007825770810311955, 5.62059734705143374752639771383, 5.92384062446878245856499389964, 6.00156105197476871734429072603, 6.07087848203495058771220942977, 6.46767298542277838367340735636, 6.74941015899702364386363172714, 6.98818305086864026162047085561, 6.98993014996485522428704601769, 7.80314647125066786448993389807

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