↑

Properties

Label 8-6825e4-1.1-c1e4-0-5
Degree $8$
Conductor $2.170\times 10^{15}$
Sign $1$
Analytic cond. $8.82102\times 10^{6}$
Root an. cond. $7.38226$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4·3-s − 3·4-s − 4·6-s + 4·7-s − 5·8-s + 10·9-s − 4·11-s + 12·12-s − 4·13-s + 4·14-s + 5·17-s + 10·18-s + 3·19-s − 16·21-s − 4·22-s + 10·23-s + 20·24-s − 4·26-s − 20·27-s − 12·28-s − 29-s − 13·31-s + 9·32-s + 16·33-s + 5·34-s − 30·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.30·3-s − 3/2·4-s − 1.63·6-s + 1.51·7-s − 1.76·8-s + 10/3·9-s − 1.20·11-s + 3.46·12-s − 1.10·13-s + 1.06·14-s + 1.21·17-s + 2.35·18-s + 0.688·19-s − 3.49·21-s − 0.852·22-s + 2.08·23-s + 4.08·24-s − 0.784·26-s − 3.84·27-s − 2.26·28-s − 0.185·29-s − 2.33·31-s + 1.59·32-s + 2.78·33-s + 0.857·34-s − 5·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(8.82102\times 10^{6}\)
Root analytic conductor: \(7.38226\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3$C_1$ \( ( 1 + T )^{4} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
13$C_1$ \( ( 1 + T )^{4} \)
good2$C_4\times C_2$ \( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 35 T^{2} + 106 T^{3} + 559 T^{4} + 106 p T^{5} + 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 5 T + 58 T^{2} - 205 T^{3} + 1389 T^{4} - 205 p T^{5} + 58 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 - 3 T + 75 T^{2} - 168 T^{3} + 2129 T^{4} - 168 p T^{5} + 75 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
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$((C_8 : C_2):C_2):C_2$ \( 1 + T + 52 T^{2} + 53 T^{3} + 2325 T^{4} + 53 p T^{5} + 52 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 13 T + 138 T^{2} + 881 T^{3} + 5705 T^{4} + 881 p T^{5} + 138 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 102 T^{2} + 565 T^{3} + 5141 T^{4} + 565 p T^{5} + 102 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 28 T^{2} + 141 T^{3} - 1125 T^{4} + 141 p T^{5} + 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 141 T^{2} - 1214 T^{3} + 7859 T^{4} - 1214 p T^{5} + 141 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 109 T^{2} + 6 p T^{3} + 5259 T^{4} + 6 p^{2} T^{5} + 109 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 11 T + 178 T^{2} + 1105 T^{3} + 11811 T^{4} + 1105 p T^{5} + 178 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 86 T^{2} + 185 T^{3} + 6181 T^{4} + 185 p T^{5} + 86 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 120 T^{2} + 639 T^{3} + 7619 T^{4} + 639 p T^{5} + 120 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 217 T^{2} + 530 T^{3} + 20011 T^{4} + 530 p T^{5} + 217 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 16 T + 350 T^{2} + 3424 T^{3} + 39439 T^{4} + 3424 p T^{5} + 350 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 16 T + 258 T^{2} - 2825 T^{3} + 28901 T^{4} - 2825 p T^{5} + 258 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 27 T + 465 T^{2} + 5682 T^{3} + 57389 T^{4} + 5682 p T^{5} + 465 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 316 T^{2} - 451 T^{3} + 38649 T^{4} - 451 p T^{5} + 316 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 436 T^{2} + 5275 T^{3} + 61461 T^{4} + 5275 p T^{5} + 436 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 262 T^{2} - 2965 T^{3} + 37261 T^{4} - 2965 p T^{5} + 262 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.79248770140057222969925364743, −5.53986311627999629662214110119, −5.52521449259148002848124275034, −5.32125666352341598971153015108, −5.25056381529396357290151953272, −4.99335480683134369434490725072, −4.92726468323214217722454340890, −4.79407051003703029363252098592, −4.75382849658011740904819178340, −4.30693626585654066652303744423, −4.26368772567159006480475619518, −4.01119235264500462862553623149, −3.89363656216571331322275921244, −3.63907029456589636841492635497, −3.23291499221777771424135895886, −3.18151675032101156652425061854, −2.98487186551236807441138553407, −2.60156038789929129104696910278, −2.42899296080476573033806290801, −2.24401554601710179919212527452, −1.76069512387483636587566545243, −1.48719003254920533118762904645, −1.24775941535274215858706161907, −1.14151817625196747251912294206, −0.994940787717938863370441638854, 0, 0, 0, 0, 0.994940787717938863370441638854, 1.14151817625196747251912294206, 1.24775941535274215858706161907, 1.48719003254920533118762904645, 1.76069512387483636587566545243, 2.24401554601710179919212527452, 2.42899296080476573033806290801, 2.60156038789929129104696910278, 2.98487186551236807441138553407, 3.18151675032101156652425061854, 3.23291499221777771424135895886, 3.63907029456589636841492635497, 3.89363656216571331322275921244, 4.01119235264500462862553623149, 4.26368772567159006480475619518, 4.30693626585654066652303744423, 4.75382849658011740904819178340, 4.79407051003703029363252098592, 4.92726468323214217722454340890, 4.99335480683134369434490725072, 5.25056381529396357290151953272, 5.32125666352341598971153015108, 5.52521449259148002848124275034, 5.53986311627999629662214110119, 5.79248770140057222969925364743

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