Properties

Degree $16$
Conductor $7.001\times 10^{26}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 14·25-s + 8·37-s − 28·43-s + 18·49-s + 40·67-s − 20·79-s + 104·109-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 40·169-s + 173-s + 56·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1.51·7-s + 14/5·25-s + 1.31·37-s − 4.26·43-s + 18/7·49-s + 4.88·67-s − 2.25·79-s + 9.96·109-s − 0.727·121-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 − 3.07·169-s + 0.0760·173-s + 4.23·175-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2268} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3240909587\)
\(L(\frac12)\) \(\approx\) \(0.3240909587\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
good5 \( ( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 4 T^{2} - 105 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 20 T^{2} + 231 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 26 T^{2} + 147 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 40 T^{2} + 759 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 8 T^{2} - 897 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - T + p T^{2} )^{8} \)
41 \( ( 1 - 79 T^{2} + 4560 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 53 T^{2} + 600 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - p T^{2} )^{8} \)
59 \( ( 1 - 43 T^{2} - 1632 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 116 T^{2} + 9735 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \)
79 \( ( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 19 T^{2} - 6528 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + 188 T^{2} + 25935 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.70327274258445749065813927654, −3.56459271092644804341032698106, −3.51698684184372321639618623884, −3.49127029445457978203840416712, −3.28258263064797625831044655545, −3.25199121856420309671050234511, −2.96741226093146595557565373121, −2.87823328155389908151593711960, −2.81621323096548305166420930373, −2.70044341204323468733776347536, −2.49359136042701177376134405642, −2.37063616686154375997038311165, −2.16216586667196662418533876146, −2.04167612002161852574529267622, −2.01260358895977304152859967400, −1.87994450247570430619104476806, −1.84333491368649619061893591297, −1.46078960532197698852893184609, −1.18291154028982388960686095866, −1.11892200331954346720229456668, −1.04332160180897363888189693860, −0.989537306630174623499633875489, −0.833529948232097098051280884403, −0.37670627136664649667685616976, −0.04265534032201478739561332983, 0.04265534032201478739561332983, 0.37670627136664649667685616976, 0.833529948232097098051280884403, 0.989537306630174623499633875489, 1.04332160180897363888189693860, 1.11892200331954346720229456668, 1.18291154028982388960686095866, 1.46078960532197698852893184609, 1.84333491368649619061893591297, 1.87994450247570430619104476806, 2.01260358895977304152859967400, 2.04167612002161852574529267622, 2.16216586667196662418533876146, 2.37063616686154375997038311165, 2.49359136042701177376134405642, 2.70044341204323468733776347536, 2.81621323096548305166420930373, 2.87823328155389908151593711960, 2.96741226093146595557565373121, 3.25199121856420309671050234511, 3.28258263064797625831044655545, 3.49127029445457978203840416712, 3.51698684184372321639618623884, 3.56459271092644804341032698106, 3.70327274258445749065813927654

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

Plot not available for L-functions of degree greater than 10.