Properties

Label 8-12e12-1.1-c1e4-0-0
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $36247.9$
Root an. cond. $3.71458$
Motivic weight $1$
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  − 24·23-s − 14·25-s − 48·47-s + 26·49-s − 24·71-s − 44·73-s − 4·97-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 5.00·23-s − 2.79·25-s − 7.00·47-s + 26/7·49-s − 2.84·71-s − 5.14·73-s − 0.406·97-s + 3.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(36247.9\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.08462769507\)
\(L(\frac12)\) \(\approx\) \(0.08462769507\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 91 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
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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

−6.63092127781956000248992701877, −6.28221150030478400795335250975, −6.19763012867478727038642270143, −5.81002754221382374083848972643, −5.79848482383253121596285957859, −5.73701106705474931150191615411, −5.67103569552439994759130649074, −5.14419398934022569791001904672, −4.70070923036105246900886340497, −4.63202250396249405435860001655, −4.51749906708880474454717611536, −4.03699460494201448611630008863, −4.02848039494135426890340102316, −3.80687117065676086238953997783, −3.71526607219495216491817544712, −3.00339614802779067589134811172, −2.99203648479962471585040335922, −2.97619489485032669404603120538, −2.27849469510758873869598817303, −1.98899400311522764530604180522, −1.73154959808209605990010538597, −1.69087824204628229877252079194, −1.55346572226865670307200775548, −0.50963717960552900332510085492, −0.07510878724780673828280787066, 0.07510878724780673828280787066, 0.50963717960552900332510085492, 1.55346572226865670307200775548, 1.69087824204628229877252079194, 1.73154959808209605990010538597, 1.98899400311522764530604180522, 2.27849469510758873869598817303, 2.97619489485032669404603120538, 2.99203648479962471585040335922, 3.00339614802779067589134811172, 3.71526607219495216491817544712, 3.80687117065676086238953997783, 4.02848039494135426890340102316, 4.03699460494201448611630008863, 4.51749906708880474454717611536, 4.63202250396249405435860001655, 4.70070923036105246900886340497, 5.14419398934022569791001904672, 5.67103569552439994759130649074, 5.73701106705474931150191615411, 5.79848482383253121596285957859, 5.81002754221382374083848972643, 6.19763012867478727038642270143, 6.28221150030478400795335250975, 6.63092127781956000248992701877

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