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Properties

Label 8-538e4-1.1-c1e4-0-0
Degree $8$
Conductor $83777829136$
Sign $1$
Analytic cond. $340.594$
Root an. cond. $2.07266$
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 + 3·3-s + 10·4-s + 5·5-s − 12·6-s − 7-s − 20·8-s − 20·10-s + 7·11-s + 30·12-s + 4·13-s + 4·14-s + 15·15-s + 35·16-s + 4·17-s + 2·19-s + 50·20-s − 3·21-s − 28·22-s + 16·23-s − 60·24-s + 2·25-s − 16·26-s − 10·27-s − 10·28-s − 60·30-s − 31-s + ⋯
L(s)  = 1  − 2.82·2-s + 1.73·3-s + 5·4-s + 2.23·5-s − 4.89·6-s − 0.377·7-s − 7.07·8-s − 6.32·10-s + 2.11·11-s + 8.66·12-s + 1.10·13-s + 1.06·14-s + 3.87·15-s + 35/4·16-s + 0.970·17-s + 0.458·19-s + 11.1·20-s − 0.654·21-s − 5.96·22-s + 3.33·23-s − 12.2·24-s + 2/5·25-s − 3.13·26-s − 1.92·27-s − 1.88·28-s − 10.9·30-s − 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 269^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 269^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 269^{4}\)
Sign: $1$
Analytic conductor: \(340.594\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 269^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.885659769\)
\(L(\frac12)\) \(\approx\) \(2.885659769\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
269$C_1$ \( ( 1 - T )^{4} \)
good3$C_2^3: C_4$ \( 1 - p T + p^{2} T^{2} - 17 T^{3} + 32 T^{4} - 17 p T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$((C_8 : C_2):C_2):C_2$ \( 1 - p T + 23 T^{2} - 69 T^{3} + 176 T^{4} - 69 p T^{5} + 23 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 + T + 22 T^{2} + 20 T^{3} + 211 T^{4} + 20 p T^{5} + 22 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 - 7 T + 56 T^{2} - 228 T^{3} + 977 T^{4} - 228 p T^{5} + 56 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 41 T^{2} - 126 T^{3} + 60 p T^{4} - 126 p T^{5} + 41 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 23 T^{2} - 72 T^{3} + 188 T^{4} - 72 p T^{5} + 23 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 52 T^{2} - 106 T^{3} + 1270 T^{4} - 106 p T^{5} + 52 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 16 T + 171 T^{2} - 1224 T^{3} + 6860 T^{4} - 1224 p T^{5} + 171 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 65 T^{2} - 136 T^{3} + 2105 T^{4} - 136 p T^{5} + 65 p^{2} T^{6} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + T + 84 T^{2} + 194 T^{3} + 3219 T^{4} + 194 p T^{5} + 84 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 107 T^{2} + 44 T^{3} + 4992 T^{4} + 44 p T^{5} + 107 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + T + 141 T^{2} + 105 T^{3} + 8252 T^{4} + 105 p T^{5} + 141 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 9 T + 94 T^{2} - 704 T^{3} + 6323 T^{4} - 704 p T^{5} + 94 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 177 T^{2} - 1285 T^{3} + 11048 T^{4} - 1285 p T^{5} + 177 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 19 T + 297 T^{2} - 3049 T^{3} + 27896 T^{4} - 3049 p T^{5} + 297 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 377 T^{2} + 3990 T^{3} + 38820 T^{4} + 3990 p T^{5} + 377 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 346 T^{2} - 3176 T^{3} + 37487 T^{4} - 3176 p T^{5} + 346 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 243 T^{2} - 2285 T^{3} + 22196 T^{4} - 2285 p T^{5} + 243 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 7 T + 134 T^{2} + 918 T^{3} + 15387 T^{4} + 918 p T^{5} + 134 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 17 T + 333 T^{2} + 3859 T^{3} + 40200 T^{4} + 3859 p T^{5} + 333 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 48 T^{2} + 490 T^{3} - 4786 T^{4} + 490 p T^{5} + 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 20 T + 421 T^{2} - 5296 T^{3} + 58905 T^{4} - 5296 p T^{5} + 421 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 3 T + 45 T^{2} - 183 T^{3} + 17108 T^{4} - 183 p T^{5} + 45 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.200609362081373821187882845746, −7.62246017138901322808141009128, −7.41173367901010032285109530688, −7.18430664689741418114004597618, −7.12031803104349283703542709383, −6.63178167518341311723693321948, −6.34434855145336749904464772859, −6.30196371455321299080964928772, −6.29566517806229637454684326185, −5.60533450717780083541617327027, −5.51794890639728733621034819689, −5.38603252494588016042016578685, −5.18577238747919645306811178745, −4.38730406642955465156931706862, −3.80936701963792566063123451023, −3.68149435561549818431328879875, −3.56124269324000870517423657380, −3.09831440209376624748036504763, −2.61259424255856168354097745297, −2.50292005003226576854470258159, −2.48415845756228362038085488963, −1.71212161812236731954651064958, −1.58602000741028475136762805962, −1.02463146886105758623892274804, −0.963312960833360907707787015655, 0.963312960833360907707787015655, 1.02463146886105758623892274804, 1.58602000741028475136762805962, 1.71212161812236731954651064958, 2.48415845756228362038085488963, 2.50292005003226576854470258159, 2.61259424255856168354097745297, 3.09831440209376624748036504763, 3.56124269324000870517423657380, 3.68149435561549818431328879875, 3.80936701963792566063123451023, 4.38730406642955465156931706862, 5.18577238747919645306811178745, 5.38603252494588016042016578685, 5.51794890639728733621034819689, 5.60533450717780083541617327027, 6.29566517806229637454684326185, 6.30196371455321299080964928772, 6.34434855145336749904464772859, 6.63178167518341311723693321948, 7.12031803104349283703542709383, 7.18430664689741418114004597618, 7.41173367901010032285109530688, 7.62246017138901322808141009128, 8.200609362081373821187882845746

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