Properties

Degree $8$
Conductor $2.252\times 10^{12}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s + 16-s + 4·71-s − 2·81-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 4·11-s + 16-s + 4·71-s − 2·81-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5375832316\)
\(L(\frac12)\) \(\approx\) \(0.5375832316\)
\(L(1)\) 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)$
bad5 \( 1 \)
7 \( 1 \)
good2$C_2^3$ \( 1 - T^{4} + T^{8} \)
3$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T + T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2^3$ \( 1 - T^{4} + T^{8} \)
29$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^3$ \( 1 - T^{4} + T^{8} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^3$ \( 1 - T^{4} + T^{8} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2^3$ \( 1 - T^{4} + T^{8} \)
71$C_2$ \( ( 1 - T + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−7.12911639013463136000110295451, −6.95859991922082585241453399050, −6.91240748978798720574315366460, −6.59709413734572691190127769182, −6.06864411573423634846182634193, −5.89244075884852514837005178803, −5.84404067243340465214747941895, −5.70432737776668382567047966735, −5.20173850773963351762683898968, −5.15318730674635754156384285009, −5.14265728771490066099987508288, −4.68077485908352453628953452659, −4.60297917776591991362979888643, −4.28739158381938449392032060958, −3.87326402879235251066294548054, −3.41331898879863638145969759964, −3.39516802043105116080282867516, −3.33972925965622599452415660909, −2.64888801856488417467285598299, −2.49933563134259233916872440427, −2.37978861687901926380447512796, −2.28612800923728337838285302019, −1.53842539097858420689547349060, −1.26785764692904594031077648994, −0.50897020455499073638655761821, 0.50897020455499073638655761821, 1.26785764692904594031077648994, 1.53842539097858420689547349060, 2.28612800923728337838285302019, 2.37978861687901926380447512796, 2.49933563134259233916872440427, 2.64888801856488417467285598299, 3.33972925965622599452415660909, 3.39516802043105116080282867516, 3.41331898879863638145969759964, 3.87326402879235251066294548054, 4.28739158381938449392032060958, 4.60297917776591991362979888643, 4.68077485908352453628953452659, 5.14265728771490066099987508288, 5.15318730674635754156384285009, 5.20173850773963351762683898968, 5.70432737776668382567047966735, 5.84404067243340465214747941895, 5.89244075884852514837005178803, 6.06864411573423634846182634193, 6.59709413734572691190127769182, 6.91240748978798720574315366460, 6.95859991922082585241453399050, 7.12911639013463136000110295451

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