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Properties

Label 8-1875e4-1.1-c1e4-0-5
Degree $8$
Conductor $1.236\times 10^{13}$
Sign $1$
Analytic cond. $50247.3$
Root an. cond. $3.86936$
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  − 2-s + 4·3-s − 3·4-s − 4·6-s − 5·7-s + 5·8-s + 10·9-s − 6·11-s − 12·12-s − 7·13-s + 5·14-s + 7·17-s − 10·18-s − 9·19-s − 20·21-s + 6·22-s − 10·23-s + 20·24-s + 7·26-s + 20·27-s + 15·28-s − 28·29-s − 10·31-s − 9·32-s − 24·33-s − 7·34-s − 30·36-s + ⋯
L(s)  = 1  − 0.707·2-s + 2.30·3-s − 3/2·4-s − 1.63·6-s − 1.88·7-s + 1.76·8-s + 10/3·9-s − 1.80·11-s − 3.46·12-s − 1.94·13-s + 1.33·14-s + 1.69·17-s − 2.35·18-s − 2.06·19-s − 4.36·21-s + 1.27·22-s − 2.08·23-s + 4.08·24-s + 1.37·26-s + 3.84·27-s + 2.83·28-s − 5.19·29-s − 1.79·31-s − 1.59·32-s − 4.17·33-s − 1.20·34-s − 5·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(50247.3\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
good2$C_4\times C_2$ \( 1 + T + p^{2} T^{2} + p T^{3} + 9 T^{4} + p^{2} T^{5} + p^{4} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 4 p T^{2} + 95 T^{3} + 289 T^{4} + 95 p T^{5} + 4 p^{3} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 50 T^{2} + 189 T^{3} + 849 T^{4} + 189 p T^{5} + 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 7 T + 46 T^{2} + 181 T^{3} + 769 T^{4} + 181 p T^{5} + 46 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 7 T + 52 T^{2} - 155 T^{3} + 831 T^{4} - 155 p T^{5} + 52 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 82 T^{2} + 477 T^{3} + 2385 T^{4} + 477 p T^{5} + 82 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 112 T^{2} + 680 T^{3} + 4089 T^{4} + 680 p T^{5} + 112 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 28 T + 400 T^{2} + 3653 T^{3} + 23319 T^{4} + 3653 p T^{5} + 400 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 4 p T^{2} + 805 T^{3} + 5641 T^{4} + 805 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 58 T^{2} - 85 T^{3} + 79 T^{4} - 85 p T^{5} + 58 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 94 T^{2} - 135 T^{3} + 4491 T^{4} - 135 p T^{5} + 94 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + T + 93 T^{2} + 470 T^{3} + 3881 T^{4} + 470 p T^{5} + 93 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 23 T + 277 T^{2} - 2230 T^{3} + 15531 T^{4} - 2230 p T^{5} + 277 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 67 T^{2} + 270 T^{3} + 4479 T^{4} + 270 p T^{5} + 67 p^{2} T^{6} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 82 T^{2} - 287 T^{3} + 4245 T^{4} - 287 p T^{5} + 82 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 43 T + 913 T^{2} + 12286 T^{3} + 114205 T^{4} + 12286 p T^{5} + 913 p^{2} T^{6} + 43 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 207 T^{2} + 1150 T^{3} + 18911 T^{4} + 1150 p T^{5} + 207 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 27 T + 418 T^{2} + 5079 T^{3} + 49545 T^{4} + 5079 p T^{5} + 418 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 15 T + 232 T^{2} + 2385 T^{3} + 25959 T^{4} + 2385 p T^{5} + 232 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 211 T^{2} - 1000 T^{3} + 17701 T^{4} - 1000 p T^{5} + 211 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 86 T^{2} + 549 T^{3} + 13989 T^{4} + 549 p T^{5} + 86 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 352 T^{2} + 2307 T^{3} + 46875 T^{4} + 2307 p T^{5} + 352 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 13 T + 172 T^{2} + 2335 T^{3} + 29251 T^{4} + 2335 p T^{5} + 172 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.25603904354815584195571563661, −7.09106401508749150117503045913, −6.63952179910422434332178837841, −6.31781397284997921234492339817, −6.13099575533624999463334293578, −5.86562372703639223268436651817, −5.61977050775997674409033349470, −5.54354549028654799920121937230, −5.49587397555262134081745935286, −4.85362078670601144020044782698, −4.65576312920894079532145534833, −4.62159502210878517638389378135, −4.28309676318721839657695599066, −4.04638820664507227371086457409, −3.79771701275218674543732389836, −3.70022471085653833959662664877, −3.63432428239703359984648049004, −3.02169601939295454506829010473, −2.98847909718467952667228511391, −2.54333964058371075751698656514, −2.41596493930272092690263645912, −2.41347399843779660794796873138, −1.68715887675010045552248791215, −1.56517115121248049417204700700, −1.52555338658801303556852893406, 0, 0, 0, 0, 1.52555338658801303556852893406, 1.56517115121248049417204700700, 1.68715887675010045552248791215, 2.41347399843779660794796873138, 2.41596493930272092690263645912, 2.54333964058371075751698656514, 2.98847909718467952667228511391, 3.02169601939295454506829010473, 3.63432428239703359984648049004, 3.70022471085653833959662664877, 3.79771701275218674543732389836, 4.04638820664507227371086457409, 4.28309676318721839657695599066, 4.62159502210878517638389378135, 4.65576312920894079532145534833, 4.85362078670601144020044782698, 5.49587397555262134081745935286, 5.54354549028654799920121937230, 5.61977050775997674409033349470, 5.86562372703639223268436651817, 6.13099575533624999463334293578, 6.31781397284997921234492339817, 6.63952179910422434332178837841, 7.09106401508749150117503045913, 7.25603904354815584195571563661

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