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Properties

Label 8-4018e4-1.1-c1e4-0-3
Degree $8$
Conductor $2.606\times 10^{14}$
Sign $1$
Analytic cond. $1.05961\times 10^{6}$
Root an. cond. $5.66426$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3·3-s + 10·4-s − 3·5-s − 12·6-s + 20·8-s + 3·9-s − 12·10-s − 30·12-s − 10·13-s + 9·15-s + 35·16-s + 17-s + 12·18-s − 14·19-s − 30·20-s − 4·23-s − 60·24-s − 7·25-s − 40·26-s − 2·27-s − 3·29-s + 36·30-s − 3·31-s + 56·32-s + 4·34-s + 30·36-s + ⋯
L(s)  = 1  + 2.82·2-s − 1.73·3-s + 5·4-s − 1.34·5-s − 4.89·6-s + 7.07·8-s + 9-s − 3.79·10-s − 8.66·12-s − 2.77·13-s + 2.32·15-s + 35/4·16-s + 0.242·17-s + 2.82·18-s − 3.21·19-s − 6.70·20-s − 0.834·23-s − 12.2·24-s − 7/5·25-s − 7.84·26-s − 0.384·27-s − 0.557·29-s + 6.57·30-s − 0.538·31-s + 9.89·32-s + 0.685·34-s + 5·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{8} \cdot 41^{4}\)
Sign: $1$
Analytic conductor: \(1.05961\times 10^{6}\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 7^{8} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
41$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + p T + 2 p T^{2} + 11 T^{3} + 22 T^{4} + 11 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 3 T + 16 T^{2} + 37 T^{3} + 116 T^{4} + 37 p T^{5} + 16 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 24 T^{2} - 10 T^{3} + 294 T^{4} - 10 p T^{5} + 24 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 10 T + 72 T^{2} + 354 T^{3} + 1478 T^{4} + 354 p T^{5} + 72 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - T + 36 T^{2} - 115 T^{3} + 618 T^{4} - 115 p T^{5} + 36 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 14 T + 132 T^{2} + 858 T^{3} + 4310 T^{4} + 858 p T^{5} + 132 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 4 T + 52 T^{2} + 208 T^{3} + 1366 T^{4} + 208 p T^{5} + 52 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 3 T + 94 T^{2} + 259 T^{3} + 3776 T^{4} + 259 p T^{5} + 94 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 3 T + 102 T^{2} + 251 T^{3} + 4394 T^{4} + 251 p T^{5} + 102 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 14 T + 184 T^{2} + 1434 T^{3} + 10654 T^{4} + 1434 p T^{5} + 184 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - T + 112 T^{2} + 127 T^{3} + 5646 T^{4} + 127 p T^{5} + 112 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 2 T + 96 T^{2} - 662 T^{3} + 4158 T^{4} - 662 p T^{5} + 96 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 11 T + 112 T^{2} - 723 T^{3} + 6700 T^{4} - 723 p T^{5} + 112 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 2 T + 154 T^{2} - 560 T^{3} + 11126 T^{4} - 560 p T^{5} + 154 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 15 T + 262 T^{2} + 2421 T^{3} + 24360 T^{4} + 2421 p T^{5} + 262 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 4 T + 192 T^{2} - 618 T^{3} + 17954 T^{4} - 618 p T^{5} + 192 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 15 T + 340 T^{2} - 3251 T^{3} + 38214 T^{4} - 3251 p T^{5} + 340 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 4 T + 228 T^{2} + 652 T^{3} + 23142 T^{4} + 652 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 15 T + 292 T^{2} - 3043 T^{3} + 34422 T^{4} - 3043 p T^{5} + 292 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 12 T + 286 T^{2} + 2126 T^{3} + 31506 T^{4} + 2126 p T^{5} + 286 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 35 T + 790 T^{2} + 11589 T^{3} + 128770 T^{4} + 11589 p T^{5} + 790 p^{2} T^{6} + 35 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 5 T + 202 T^{2} - 555 T^{3} + 24202 T^{4} - 555 p T^{5} + 202 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.35664994178247612274208320519, −5.90923400669251891520754163609, −5.78893404958645532620462129482, −5.69964837897626235392383163332, −5.54251458816339344850619639131, −5.23444232669738771616609091919, −5.21945941576805216121509630676, −5.14118583812174923711477429135, −4.81437210724293323826717393740, −4.41511995020011893065545049431, −4.34240004878116253046864846469, −4.23881712565802281591183138497, −4.14440211500312221512282078661, −3.87546054119031663089079993301, −3.67472978436585810427418555919, −3.57131987621155125751375573583, −3.26063138750942650815820701551, −2.75405553266864708488598466433, −2.71245273900437060553579855274, −2.43962137553585885858118387810, −2.25414520640031099705135027118, −1.98247498361068665404075164520, −1.83832732495287979125839646767, −1.44774078910000985340002766400, −1.12562112955728722704760341401, 0, 0, 0, 0, 1.12562112955728722704760341401, 1.44774078910000985340002766400, 1.83832732495287979125839646767, 1.98247498361068665404075164520, 2.25414520640031099705135027118, 2.43962137553585885858118387810, 2.71245273900437060553579855274, 2.75405553266864708488598466433, 3.26063138750942650815820701551, 3.57131987621155125751375573583, 3.67472978436585810427418555919, 3.87546054119031663089079993301, 4.14440211500312221512282078661, 4.23881712565802281591183138497, 4.34240004878116253046864846469, 4.41511995020011893065545049431, 4.81437210724293323826717393740, 5.14118583812174923711477429135, 5.21945941576805216121509630676, 5.23444232669738771616609091919, 5.54251458816339344850619639131, 5.69964837897626235392383163332, 5.78893404958645532620462129482, 5.90923400669251891520754163609, 6.35664994178247612274208320519

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