Properties

Label 8-4598e4-1.1-c1e4-0-3
Degree $8$
Conductor $4.470\times 10^{14}$
Sign $1$
Analytic cond. $1.81712\times 10^{6}$
Root an. cond. $6.05930$
Motivic weight $1$
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  − 4·2-s + 3-s + 10·4-s + 3·5-s − 4·6-s + 7-s − 20·8-s − 3·9-s − 12·10-s + 10·12-s + 9·13-s − 4·14-s + 3·15-s + 35·16-s + 2·17-s + 12·18-s + 4·19-s + 30·20-s + 21-s − 2·23-s − 20·24-s + 2·25-s − 36·26-s − 5·27-s + 10·28-s − 5·29-s − 12·30-s + ⋯
L(s)  = 1  − 2.82·2-s + 0.577·3-s + 5·4-s + 1.34·5-s − 1.63·6-s + 0.377·7-s − 7.07·8-s − 9-s − 3.79·10-s + 2.88·12-s + 2.49·13-s − 1.06·14-s + 0.774·15-s + 35/4·16-s + 0.485·17-s + 2.82·18-s + 0.917·19-s + 6.70·20-s + 0.218·21-s − 0.417·23-s − 4.08·24-s + 2/5·25-s − 7.06·26-s − 0.962·27-s + 1.88·28-s − 0.928·29-s − 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 11^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.81712\times 10^{6}\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4598} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 11^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.804747215\)
\(L(\frac12)\) \(\approx\) \(4.804747215\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
19$C_1$ \( ( 1 - T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - T + 4 T^{2} - 2 T^{3} + 5 T^{4} - 2 p T^{5} + 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 3 T + 7 T^{2} - 9 T^{3} + 16 T^{4} - 9 p T^{5} + 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - T + 18 T^{2} - 2 T^{3} + 3 p^{2} T^{4} - 2 p T^{5} + 18 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 9 T + 66 T^{2} - 336 T^{3} + 105 p T^{4} - 336 p T^{5} + 66 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 2 T + 49 T^{2} - 64 T^{3} + 1116 T^{4} - 64 p T^{5} + 49 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 2 T + 77 T^{2} + 118 T^{3} + 2488 T^{4} + 118 p T^{5} + 77 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 5 T + 30 T^{2} + 216 T^{3} + 2081 T^{4} + 216 p T^{5} + 30 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 27 T + 373 T^{2} - 3367 T^{3} + 21848 T^{4} - 3367 p T^{5} + 373 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 6 T + 80 T^{2} - 442 T^{3} + 3054 T^{4} - 442 p T^{5} + 80 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 5 T + 91 T^{2} - 463 T^{3} + 4896 T^{4} - 463 p T^{5} + 91 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + T + 107 T^{2} - 179 T^{3} + 5112 T^{4} - 179 p T^{5} + 107 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 22 T + 304 T^{2} - 2942 T^{3} + 22750 T^{4} - 2942 p T^{5} + 304 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 105 T^{2} + 564 T^{3} + 4728 T^{4} + 564 p T^{5} + 105 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 157 T^{2} + 72 T^{3} + 12652 T^{4} + 72 p T^{5} + 157 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 6 T + 216 T^{2} + 18 p T^{3} + 18942 T^{4} + 18 p^{2} T^{5} + 216 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 17 T + 326 T^{2} - 3450 T^{3} + 34683 T^{4} - 3450 p T^{5} + 326 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 19 T + 387 T^{2} + 4167 T^{3} + 44900 T^{4} + 4167 p T^{5} + 387 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 20 T + 413 T^{2} + 4592 T^{3} + 49752 T^{4} + 4592 p T^{5} + 413 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 12 T + 208 T^{2} + 1324 T^{3} + 17918 T^{4} + 1324 p T^{5} + 208 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 23 T + 375 T^{2} - 4591 T^{3} + 44424 T^{4} - 4591 p T^{5} + 375 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 16 T + 288 T^{2} - 2816 T^{3} + 34686 T^{4} - 2816 p T^{5} + 288 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 8 T + 172 T^{2} - 1720 T^{3} + 25174 T^{4} - 1720 p T^{5} + 172 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.05977360400240271099325508969, −5.84032017611929325412434763767, −5.64153436616335667169365166609, −5.49333981594572747888659538258, −5.44522768966150804007334306952, −4.83231541950650333474401526495, −4.72990298617147611073059594334, −4.41851643715058228194144217155, −4.36282948946776859234760809585, −4.04779848926035873181339456592, −3.59530387491869567977843992578, −3.49937122741865751543156669854, −3.26828654519681593572344130523, −3.05301460317335736327961977033, −2.79680450110051065898305431276, −2.65341685761541184894731410214, −2.58706125201397096783267813807, −2.01133049881037536941929459089, −1.96566334169124693427896503944, −1.79581628320343702783251276045, −1.38484509504537620049637323623, −1.22976112593117943035176863945, −0.76812334043451348493565801371, −0.75278835288491543619475530899, −0.54512057960230178295176708463, 0.54512057960230178295176708463, 0.75278835288491543619475530899, 0.76812334043451348493565801371, 1.22976112593117943035176863945, 1.38484509504537620049637323623, 1.79581628320343702783251276045, 1.96566334169124693427896503944, 2.01133049881037536941929459089, 2.58706125201397096783267813807, 2.65341685761541184894731410214, 2.79680450110051065898305431276, 3.05301460317335736327961977033, 3.26828654519681593572344130523, 3.49937122741865751543156669854, 3.59530387491869567977843992578, 4.04779848926035873181339456592, 4.36282948946776859234760809585, 4.41851643715058228194144217155, 4.72990298617147611073059594334, 4.83231541950650333474401526495, 5.44522768966150804007334306952, 5.49333981594572747888659538258, 5.64153436616335667169365166609, 5.84032017611929325412434763767, 6.05977360400240271099325508969

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