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Properties

Label 8-150e4-1.1-c13e4-0-6
Degree $8$
Conductor $506250000$
Sign $1$
Analytic cond. $6.69337\times 10^{8}$
Root an. cond. $12.6825$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.19e3·4-s − 1.06e6·9-s + 8.03e6·11-s + 5.03e7·16-s + 1.81e8·19-s − 4.14e9·29-s + 2.34e10·31-s + 8.70e9·36-s + 1.35e11·41-s − 6.58e10·44-s + 1.93e11·49-s + 8.66e9·59-s + 1.77e12·61-s − 2.74e11·64-s + 1.87e12·71-s − 1.48e12·76-s + 2.49e12·79-s + 8.47e11·81-s + 1.60e13·89-s − 8.54e12·99-s + 1.54e13·101-s − 6.36e13·109-s + 3.39e13·116-s + 3.27e13·121-s − 1.92e14·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s − 2/3·9-s + 1.36·11-s + 3/4·16-s + 0.883·19-s − 1.29·29-s + 4.74·31-s + 2/3·36-s + 4.46·41-s − 1.36·44-s + 1.99·49-s + 0.0267·59-s + 4.41·61-s − 1/2·64-s + 1.73·71-s − 0.883·76-s + 1.15·79-s + 1/3·81-s + 3.41·89-s − 0.911·99-s + 1.45·101-s − 3.63·109-s + 1.29·116-s + 0.949·121-s − 4.74·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(6.69337\times 10^{8}\)
Root analytic conductor: \(12.6825\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 13/2, 13/2, 13/2, 13/2 ),\ 1 )\)

Particular Values

\(L(7)\) \(\approx\) \(15.97682924\)
\(L(\frac12)\) \(\approx\) \(15.97682924\)
\(L(\frac{15}{2})\) 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$C_2$ \( ( 1 + p^{12} T^{2} )^{2} \)
3$C_2$ \( ( 1 + p^{12} T^{2} )^{2} \)
5 \( 1 \)
good7$D_4\times C_2$ \( 1 - 193061448956 T^{2} + \)\(47\!\cdots\!18\)\( p^{2} T^{4} - 193061448956 p^{26} T^{6} + p^{52} T^{8} \)
11$D_{4}$ \( ( 1 - 4018560 T + 712881822242 p T^{2} - 4018560 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 256375221382516 T^{2} + \)\(76\!\cdots\!82\)\( T^{4} + 256375221382516 p^{26} T^{6} + p^{52} T^{8} \)
17$D_4\times C_2$ \( 1 - 27320361335013596 T^{2} + \)\(37\!\cdots\!42\)\( T^{4} - 27320361335013596 p^{26} T^{6} + p^{52} T^{8} \)
19$D_{4}$ \( ( 1 - 90569648 T + 22422218733426 p T^{2} - 90569648 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 1113072945396229532 T^{2} + \)\(61\!\cdots\!34\)\( T^{4} - 1113072945396229532 p^{26} T^{6} + p^{52} T^{8} \)
29$D_{4}$ \( ( 1 + 2072310228 T - 1269722572062225026 T^{2} + 2072310228 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 11728406776 T + 78923255143260763326 T^{2} - 11728406776 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + \)\(20\!\cdots\!40\)\( T^{2} + \)\(82\!\cdots\!18\)\( T^{4} + \)\(20\!\cdots\!40\)\( p^{26} T^{6} + p^{52} T^{8} \)
41$D_{4}$ \( ( 1 - 67871568228 T + \)\(28\!\cdots\!38\)\( T^{2} - 67871568228 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(38\!\cdots\!80\)\( T^{2} + \)\(94\!\cdots\!98\)\( T^{4} - \)\(38\!\cdots\!80\)\( p^{26} T^{6} + p^{52} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(16\!\cdots\!08\)\( T^{2} + \)\(12\!\cdots\!74\)\( T^{4} - \)\(16\!\cdots\!08\)\( p^{26} T^{6} + p^{52} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(10\!\cdots\!80\)\( T^{2} + \)\(40\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!80\)\( p^{26} T^{6} + p^{52} T^{8} \)
59$D_{4}$ \( ( 1 - 4334346240 T + \)\(13\!\cdots\!58\)\( T^{2} - 4334346240 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 887411684860 T + \)\(51\!\cdots\!62\)\( T^{2} - 887411684860 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(16\!\cdots\!76\)\( T^{2} - \)\(58\!\cdots\!18\)\( T^{4} - \)\(16\!\cdots\!76\)\( p^{26} T^{6} + p^{52} T^{8} \)
71$D_{4}$ \( ( 1 - 936151332480 T + \)\(24\!\cdots\!22\)\( T^{2} - 936151332480 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + \)\(18\!\cdots\!36\)\( T^{2} + \)\(34\!\cdots\!02\)\( T^{4} + \)\(18\!\cdots\!36\)\( p^{26} T^{6} + p^{52} T^{8} \)
79$D_{4}$ \( ( 1 - 1248629414744 T + \)\(97\!\cdots\!62\)\( T^{2} - 1248629414744 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + \)\(20\!\cdots\!16\)\( T^{2} + \)\(15\!\cdots\!02\)\( T^{4} + \)\(20\!\cdots\!16\)\( p^{26} T^{6} + p^{52} T^{8} \)
89$D_{4}$ \( ( 1 - 8016259169484 T + \)\(56\!\cdots\!02\)\( T^{2} - 8016259169484 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!80\)\( T^{2} + \)\(14\!\cdots\!58\)\( T^{4} - \)\(14\!\cdots\!80\)\( p^{26} T^{6} + p^{52} T^{8} \)
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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

−6.99836686116854831564813594527, −6.93333589180756930702386681301, −6.44714494730117862283413612717, −6.43590918556170575591975112259, −5.93562286423486722540774706201, −5.72681382527298886816322047678, −5.63832124447677858504592078197, −5.24881268173533652120366704001, −4.68794780875136183851663559114, −4.66862628131449137384718606455, −4.52363319841899838801805639757, −3.93637419285747787979236832127, −3.77569399235088099196215077472, −3.68051293230362102583589705443, −3.39012100718203461098929991099, −2.71356724354052112614662965930, −2.51422471142133244880960151341, −2.34884203777837643029533101410, −2.33421456445794665362536010490, −1.46955994353252120038359559771, −1.27453790677467582891995547495, −0.803215243148571459180002067595, −0.72534603805391579287241433576, −0.69850709885191978608748800460, −0.45357001245303728305617500777, 0.45357001245303728305617500777, 0.69850709885191978608748800460, 0.72534603805391579287241433576, 0.803215243148571459180002067595, 1.27453790677467582891995547495, 1.46955994353252120038359559771, 2.33421456445794665362536010490, 2.34884203777837643029533101410, 2.51422471142133244880960151341, 2.71356724354052112614662965930, 3.39012100718203461098929991099, 3.68051293230362102583589705443, 3.77569399235088099196215077472, 3.93637419285747787979236832127, 4.52363319841899838801805639757, 4.66862628131449137384718606455, 4.68794780875136183851663559114, 5.24881268173533652120366704001, 5.63832124447677858504592078197, 5.72681382527298886816322047678, 5.93562286423486722540774706201, 6.43590918556170575591975112259, 6.44714494730117862283413612717, 6.93333589180756930702386681301, 6.99836686116854831564813594527

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