Properties

Label 8-48e8-1.1-c2e4-0-1
Degree $8$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $1.55335\times 10^{7}$
Root an. cond. $7.92334$
Motivic weight $2$
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  − 32·7-s − 96·25-s + 160·31-s + 444·49-s − 384·73-s − 416·79-s − 320·97-s + 288·103-s − 228·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 548·169-s + 173-s + 3.07e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 4.57·7-s − 3.83·25-s + 5.16·31-s + 9.06·49-s − 5.26·73-s − 5.26·79-s − 3.29·97-s + 2.79·103-s − 1.88·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.24·169-s + 0.00578·173-s + 17.5·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.55335\times 10^{7}\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0005731790346\)
\(L(\frac12)\) \(\approx\) \(0.0005731790346\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 + 48 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + 114 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 416 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 302 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 94 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 240 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 1056 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 3442 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 4290 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 4560 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 6450 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 4526 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 2578 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 3810 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 96 T + p^{2} T^{2} )^{4} \)
79$C_2$ \( ( 1 + 104 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 3410 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 9792 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 80 T + p^{2} T^{2} )^{4} \)
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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

−6.14647719220171940046115242553, −6.09077616887823631630613589797, −6.05070379627125828743973424398, −5.76904760750194826630963391705, −5.49153526887748362746881151699, −5.39816563331625518987754171549, −4.82685441565450504527570724668, −4.60965223868030921285037204972, −4.34048504619227800822847280945, −4.21997817161554947594306570506, −4.19511532519424435870255524045, −3.63843157890471106415011584868, −3.53437761784028678766360960763, −3.47850624896089051184017132241, −3.03061619013472631914891315881, −2.85974755672689855418563153033, −2.65883473144462392459839017043, −2.55977495382256667386514216578, −2.52443740819015770592049993572, −1.70451132432863807924302709996, −1.43393750777722667586831938109, −1.29518052050737481884793050177, −0.74745755211716692610772701923, −0.22209249059355366098538954002, −0.008980643404592978051554967899, 0.008980643404592978051554967899, 0.22209249059355366098538954002, 0.74745755211716692610772701923, 1.29518052050737481884793050177, 1.43393750777722667586831938109, 1.70451132432863807924302709996, 2.52443740819015770592049993572, 2.55977495382256667386514216578, 2.65883473144462392459839017043, 2.85974755672689855418563153033, 3.03061619013472631914891315881, 3.47850624896089051184017132241, 3.53437761784028678766360960763, 3.63843157890471106415011584868, 4.19511532519424435870255524045, 4.21997817161554947594306570506, 4.34048504619227800822847280945, 4.60965223868030921285037204972, 4.82685441565450504527570724668, 5.39816563331625518987754171549, 5.49153526887748362746881151699, 5.76904760750194826630963391705, 6.05070379627125828743973424398, 6.09077616887823631630613589797, 6.14647719220171940046115242553

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