Properties

Label 8-1183e4-1.1-c1e4-0-1
Degree $8$
Conductor $1.959\times 10^{12}$
Sign $1$
Analytic cond. $7962.46$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 5·4-s + 8·7-s − 6·8-s + 9-s − 6·11-s − 24·14-s + 4·16-s + 6·17-s − 3·18-s + 6·19-s + 18·22-s + 12·23-s + 5·25-s + 40·28-s + 10·31-s − 18·34-s + 5·36-s − 4·37-s − 18·38-s − 32·43-s − 30·44-s − 36·46-s − 6·47-s + 34·49-s − 15·50-s + 6·53-s + ⋯
L(s)  = 1  − 2.12·2-s + 5/2·4-s + 3.02·7-s − 2.12·8-s + 1/3·9-s − 1.80·11-s − 6.41·14-s + 16-s + 1.45·17-s − 0.707·18-s + 1.37·19-s + 3.83·22-s + 2.50·23-s + 25-s + 7.55·28-s + 1.79·31-s − 3.08·34-s + 5/6·36-s − 0.657·37-s − 2.91·38-s − 4.87·43-s − 4.52·44-s − 5.30·46-s − 0.875·47-s + 34/7·49-s − 2.12·50-s + 0.824·53-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(7^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(7962.46\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1183} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.812292977\)
\(L(\frac12)\) \(\approx\) \(1.812292977\)
\(L(\frac{3}{2})\) 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)$
bad7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13 \( 1 \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \)
3$C_2^3$ \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
5$C_2$$\times$$C_2^2$ \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 6 T + 13 T^{2} + 66 T^{3} - 372 T^{4} + 66 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 12 T + 67 T^{2} - 372 T^{3} + 2088 T^{4} - 372 p T^{5} + 67 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 4 T - 17 T^{2} - 164 T^{3} - 872 T^{4} - 164 p T^{5} - 17 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 + 6 T - p T^{2} - 66 T^{3} + 2988 T^{4} - 66 p T^{5} - p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 6 T - 59 T^{2} + 66 T^{3} + 4308 T^{4} + 66 p T^{5} - 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 6 T - 71 T^{2} + 66 T^{3} + 5844 T^{4} + 66 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 8 T - 53 T^{2} + 232 T^{3} + 3688 T^{4} + 232 p T^{5} - 53 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 8 T - 65 T^{2} - 232 T^{3} + 5344 T^{4} - 232 p T^{5} - 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^3$ \( 1 - 173 T^{2} + 22008 T^{4} - 173 p^{2} T^{6} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.20671776803264796450101768412, −6.79435386943882934250955087155, −6.75286636278374088542904287927, −6.64868455019575157540133013813, −6.12357493532839406634008462336, −5.79503195350725474188478093557, −5.71255287163473565103921145306, −5.22700521423171777629234952259, −5.03946911914954901355657644264, −4.91784962875383524269049550069, −4.90077805720536869481317503158, −4.67872993610053056427934158397, −4.65280073881114745836965557753, −3.66496200295194288997885792236, −3.61390989824593194726994036616, −3.46850859200000033925920931681, −2.79488112877270414127612213735, −2.78711187513178353260298122715, −2.70022235914273561840142761034, −1.96460054055032530093593554931, −1.78174561676838301276531874606, −1.52803282246173312726481778507, −1.17834069280827079155607748349, −0.988314660888264588742939175370, −0.46928506186351079857578087747, 0.46928506186351079857578087747, 0.988314660888264588742939175370, 1.17834069280827079155607748349, 1.52803282246173312726481778507, 1.78174561676838301276531874606, 1.96460054055032530093593554931, 2.70022235914273561840142761034, 2.78711187513178353260298122715, 2.79488112877270414127612213735, 3.46850859200000033925920931681, 3.61390989824593194726994036616, 3.66496200295194288997885792236, 4.65280073881114745836965557753, 4.67872993610053056427934158397, 4.90077805720536869481317503158, 4.91784962875383524269049550069, 5.03946911914954901355657644264, 5.22700521423171777629234952259, 5.71255287163473565103921145306, 5.79503195350725474188478093557, 6.12357493532839406634008462336, 6.64868455019575157540133013813, 6.75286636278374088542904287927, 6.79435386943882934250955087155, 7.20671776803264796450101768412

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