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Properties

Label 8-33e8-1.1-c3e4-0-3
Degree $8$
Conductor $1.406\times 10^{12}$
Sign $1$
Analytic cond. $1.70441\times 10^{7}$
Root an. cond. $8.01580$
Motivic weight $3$
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 − 4-s + 11·5-s − 25·7-s − 14·8-s + 44·10-s − 25·13-s − 100·14-s + 5·16-s + 232·17-s − 154·19-s − 11·20-s + 6·23-s − 196·25-s − 100·26-s + 25·28-s + 363·29-s + 37·31-s − 88·32-s + 928·34-s − 275·35-s + 93·37-s − 616·38-s − 154·40-s + 152·41-s + 325·43-s + 24·46-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/8·4-s + 0.983·5-s − 1.34·7-s − 0.618·8-s + 1.39·10-s − 0.533·13-s − 1.90·14-s + 5/64·16-s + 3.30·17-s − 1.85·19-s − 0.122·20-s + 0.0543·23-s − 1.56·25-s − 0.754·26-s + 0.168·28-s + 2.32·29-s + 0.214·31-s − 0.486·32-s + 4.68·34-s − 1.32·35-s + 0.413·37-s − 2.62·38-s − 0.608·40-s + 0.578·41-s + 1.15·43-s + 0.0769·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.70441\times 10^{7}\)
Root analytic conductor: \(8.01580\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(8.712791917\)
\(L(\frac12)\) \(\approx\) \(8.712791917\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11 \( 1 \)
good2$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - p^{2} T + 17 T^{2} - 29 p T^{3} + 47 p^{2} T^{4} - 29 p^{4} T^{5} + 17 p^{6} T^{6} - p^{11} T^{7} + p^{12} T^{8} \)
5$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 11 T + 317 T^{2} - 3299 T^{3} + 56036 T^{4} - 3299 p^{3} T^{5} + 317 p^{6} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} \)
7$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 25 T + 1121 T^{2} + 25005 T^{3} + 533312 T^{4} + 25005 p^{3} T^{5} + 1121 p^{6} T^{6} + 25 p^{9} T^{7} + p^{12} T^{8} \)
13$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 25 T + 6573 T^{2} + 186125 T^{3} + 19174244 T^{4} + 186125 p^{3} T^{5} + 6573 p^{6} T^{6} + 25 p^{9} T^{7} + p^{12} T^{8} \)
17$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 232 T + 37837 T^{2} - 3942852 T^{3} + 326611013 T^{4} - 3942852 p^{3} T^{5} + 37837 p^{6} T^{6} - 232 p^{9} T^{7} + p^{12} T^{8} \)
19$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 154 T + 27937 T^{2} + 2750902 T^{3} + 279655675 T^{4} + 2750902 p^{3} T^{5} + 27937 p^{6} T^{6} + 154 p^{9} T^{7} + p^{12} T^{8} \)
23$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 6 T + 27488 T^{2} - 757614 T^{3} + 422704798 T^{4} - 757614 p^{3} T^{5} + 27488 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
29$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 363 T + 119679 T^{2} - 24430329 T^{3} + 4549527256 T^{4} - 24430329 p^{3} T^{5} + 119679 p^{6} T^{6} - 363 p^{9} T^{7} + p^{12} T^{8} \)
31$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 37 T + 62039 T^{2} + 2822175 T^{3} + 1888178140 T^{4} + 2822175 p^{3} T^{5} + 62039 p^{6} T^{6} - 37 p^{9} T^{7} + p^{12} T^{8} \)
37$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 93 T + 175897 T^{2} - 12830673 T^{3} + 12851212828 T^{4} - 12830673 p^{3} T^{5} + 175897 p^{6} T^{6} - 93 p^{9} T^{7} + p^{12} T^{8} \)
41$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 152 T + 190517 T^{2} - 25342148 T^{3} + 18420120413 T^{4} - 25342148 p^{3} T^{5} + 190517 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8} \)
43$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 325 T + 241017 T^{2} - 48348645 T^{3} + 24393602404 T^{4} - 48348645 p^{3} T^{5} + 241017 p^{6} T^{6} - 325 p^{9} T^{7} + p^{12} T^{8} \)
47$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 869 T + 659213 T^{2} - 296907125 T^{3} + 116162199476 T^{4} - 296907125 p^{3} T^{5} + 659213 p^{6} T^{6} - 869 p^{9} T^{7} + p^{12} T^{8} \)
53$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 811 T + 678683 T^{2} + 344528809 T^{3} + 156904141148 T^{4} + 344528809 p^{3} T^{5} + 678683 p^{6} T^{6} + 811 p^{9} T^{7} + p^{12} T^{8} \)
59$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 178 T + 745201 T^{2} + 110327222 T^{3} + 221994353295 T^{4} + 110327222 p^{3} T^{5} + 745201 p^{6} T^{6} + 178 p^{9} T^{7} + p^{12} T^{8} \)
61$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 105 T + 360643 T^{2} + 6849045 T^{3} + 108563358588 T^{4} + 6849045 p^{3} T^{5} + 360643 p^{6} T^{6} - 105 p^{9} T^{7} + p^{12} T^{8} \)
67$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 43 T + 808331 T^{2} - 155629659 T^{3} + 296445352544 T^{4} - 155629659 p^{3} T^{5} + 808331 p^{6} T^{6} - 43 p^{9} T^{7} + p^{12} T^{8} \)
71$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 629 T + 1107589 T^{2} + 711566499 T^{3} + 535891483520 T^{4} + 711566499 p^{3} T^{5} + 1107589 p^{6} T^{6} + 629 p^{9} T^{7} + p^{12} T^{8} \)
73$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 270 T + 12543 p T^{2} - 362261860 T^{3} + 447779949957 T^{4} - 362261860 p^{3} T^{5} + 12543 p^{7} T^{6} - 270 p^{9} T^{7} + p^{12} T^{8} \)
79$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 977 T + 2039219 T^{2} + 1350401001 T^{3} + 1524443490076 T^{4} + 1350401001 p^{3} T^{5} + 2039219 p^{6} T^{6} + 977 p^{9} T^{7} + p^{12} T^{8} \)
83$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 1686 T + 2122689 T^{2} - 1844426202 T^{3} + 1592311847239 T^{4} - 1844426202 p^{3} T^{5} + 2122689 p^{6} T^{6} - 1686 p^{9} T^{7} + p^{12} T^{8} \)
89$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 1891 T + 3678143 T^{2} + 4008289261 T^{3} + 4189944324368 T^{4} + 4008289261 p^{3} T^{5} + 3678143 p^{6} T^{6} + 1891 p^{9} T^{7} + p^{12} T^{8} \)
97$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 1772 T + 3744705 T^{2} + 3890953980 T^{3} + 4812316950973 T^{4} + 3890953980 p^{3} T^{5} + 3744705 p^{6} T^{6} + 1772 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.39609381988744677330449936150, −6.12589947743891151738009105889, −6.08229772735676063508203029033, −6.07864324072281459552915154864, −5.98419903044702606078144499000, −5.40234379703421742276857712652, −5.21767946475341081870785087016, −5.05951239044436151500816054400, −4.97905692100698843420679365982, −4.40125783405203335679393722680, −4.36658705927954232858691491132, −4.28184402403416684919518701925, −3.91018952044423128522929976280, −3.57501997092227735392526382742, −3.37882968146604862615138904253, −3.27042628894040446247863773457, −2.77567051884976588864864500107, −2.56873092976594150194041567612, −2.50068521118251704243986400340, −1.91672232832010952684078884567, −1.65574694170226513342510542382, −1.32320636950909172474630138947, −0.929784282461443647646575852409, −0.50571722534609874436187903272, −0.33874504037618698211217085759, 0.33874504037618698211217085759, 0.50571722534609874436187903272, 0.929784282461443647646575852409, 1.32320636950909172474630138947, 1.65574694170226513342510542382, 1.91672232832010952684078884567, 2.50068521118251704243986400340, 2.56873092976594150194041567612, 2.77567051884976588864864500107, 3.27042628894040446247863773457, 3.37882968146604862615138904253, 3.57501997092227735392526382742, 3.91018952044423128522929976280, 4.28184402403416684919518701925, 4.36658705927954232858691491132, 4.40125783405203335679393722680, 4.97905692100698843420679365982, 5.05951239044436151500816054400, 5.21767946475341081870785087016, 5.40234379703421742276857712652, 5.98419903044702606078144499000, 6.07864324072281459552915154864, 6.08229772735676063508203029033, 6.12589947743891151738009105889, 6.39609381988744677330449936150

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