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Properties

Label 16-1110e8-1.1-c1e8-0-1
Degree $16$
Conductor $2.305\times 10^{24}$
Sign $1$
Analytic cond. $3.80890\times 10^{7}$
Root an. cond. $2.97714$
Motivic weight $1$
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  + 8·3-s − 4·4-s − 4·7-s + 36·9-s − 4·11-s − 32·12-s + 10·16-s − 32·21-s − 4·25-s + 120·27-s + 16·28-s − 32·33-s − 144·36-s + 18·37-s − 10·41-s + 16·44-s + 28·47-s + 80·48-s − 6·49-s + 12·53-s − 144·63-s − 20·64-s + 12·67-s + 24·71-s + 10·73-s − 32·75-s + 16·77-s + ⋯
L(s)  = 1  + 4.61·3-s − 2·4-s − 1.51·7-s + 12·9-s − 1.20·11-s − 9.23·12-s + 5/2·16-s − 6.98·21-s − 4/5·25-s + 23.0·27-s + 3.02·28-s − 5.57·33-s − 24·36-s + 2.95·37-s − 1.56·41-s + 2.41·44-s + 4.08·47-s + 11.5·48-s − 6/7·49-s + 1.64·53-s − 18.1·63-s − 5/2·64-s + 1.46·67-s + 2.84·71-s + 1.17·73-s − 3.69·75-s + 1.82·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}\)
Sign: $1$
Analytic conductor: \(3.80890\times 10^{7}\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(23.15347402\)
\(L(\frac12)\) \(\approx\) \(23.15347402\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
3 \( ( 1 - T )^{8} \)
5 \( ( 1 + T^{2} )^{4} \)
37 \( 1 - 18 T + 120 T^{2} - 230 T^{3} - 610 T^{4} - 230 p T^{5} + 120 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
good7 \( ( 1 + 2 T + 9 T^{2} + 6 T^{3} + 44 T^{4} + 6 p T^{5} + 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 2 T + p T^{2} + 28 T^{3} + 60 T^{4} + 28 p T^{5} + p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 51 T^{2} + 1450 T^{4} - 27925 T^{6} + 412138 T^{8} - 27925 p^{2} T^{10} + 1450 p^{4} T^{12} - 51 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 86 T^{2} + 3449 T^{4} - 88262 T^{6} + 1685524 T^{8} - 88262 p^{2} T^{10} + 3449 p^{4} T^{12} - 86 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 - 71 T^{2} + 3066 T^{4} - 90321 T^{6} + 1978538 T^{8} - 90321 p^{2} T^{10} + 3066 p^{4} T^{12} - 71 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 - 131 T^{2} + 8350 T^{4} - 334125 T^{6} + 9172738 T^{8} - 334125 p^{2} T^{10} + 8350 p^{4} T^{12} - 131 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 - 19 T^{2} + 1426 T^{4} - 9669 T^{6} + 1269306 T^{8} - 9669 p^{2} T^{10} + 1426 p^{4} T^{12} - 19 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 131 T^{2} + 8502 T^{4} - 362061 T^{6} + 12194738 T^{8} - 362061 p^{2} T^{10} + 8502 p^{4} T^{12} - 131 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 5 T + 118 T^{2} + 563 T^{3} + 6674 T^{4} + 563 p T^{5} + 118 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 255 T^{2} + 30706 T^{4} - 2298169 T^{6} + 117805018 T^{8} - 2298169 p^{2} T^{10} + 30706 p^{4} T^{12} - 255 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 - 14 T + 64 T^{2} + 586 T^{3} - 7618 T^{4} + 586 p T^{5} + 64 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 6 T + 205 T^{2} - 922 T^{3} + 16124 T^{4} - 922 p T^{5} + 205 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 260 T^{2} + 36052 T^{4} - 3325212 T^{6} + 226302646 T^{8} - 3325212 p^{2} T^{10} + 36052 p^{4} T^{12} - 260 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 311 T^{2} + 47598 T^{4} - 77205 p T^{6} + 334310210 T^{8} - 77205 p^{3} T^{10} + 47598 p^{4} T^{12} - 311 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 - 6 T + 60 T^{2} + 74 T^{3} + 3158 T^{4} + 74 p T^{5} + 60 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 12 T + 256 T^{2} - 2300 T^{3} + 26462 T^{4} - 2300 p T^{5} + 256 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 5 T + 210 T^{2} - 867 T^{3} + 21882 T^{4} - 867 p T^{5} + 210 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 - 140 T^{2} + 21204 T^{4} - 2466036 T^{6} + 185705942 T^{8} - 2466036 p^{2} T^{10} + 21204 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 - 3 T + 208 T^{2} + 5 T^{3} + 20174 T^{4} + 5 p T^{5} + 208 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 - 527 T^{2} + 132910 T^{4} - 20902113 T^{6} + 2232605602 T^{8} - 20902113 p^{2} T^{10} + 132910 p^{4} T^{12} - 527 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 275 T^{2} + 62634 T^{4} - 8629437 T^{6} + 1011478730 T^{8} - 8629437 p^{2} T^{10} + 62634 p^{4} T^{12} - 275 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.04746385607230922513384260244, −4.02547163112643252288104562762, −3.92330080889104943902022518475, −3.78038841065032231900085266698, −3.61934492578378904858909251691, −3.54711612946183544983508582897, −3.51886649165010669220867172732, −3.37390782374923872688219762636, −3.18997448682146557318369095360, −2.93937626845352026859992847508, −2.74263697392464393422584078254, −2.61108377119400778804633141053, −2.60696791974846211043663761359, −2.48200156597610685879991356126, −2.45048548525437259149616380828, −2.40818767875137690903752965027, −2.25555573660070458621494075485, −1.75279153798382622683326816128, −1.64349430186259516883569783444, −1.40684933185677598087008931822, −1.35741105328725901855168827261, −1.18847311637681584561643726167, −0.71282682693049078513423651912, −0.48672693371614329752513174304, −0.43816467838509020214220256905, 0.43816467838509020214220256905, 0.48672693371614329752513174304, 0.71282682693049078513423651912, 1.18847311637681584561643726167, 1.35741105328725901855168827261, 1.40684933185677598087008931822, 1.64349430186259516883569783444, 1.75279153798382622683326816128, 2.25555573660070458621494075485, 2.40818767875137690903752965027, 2.45048548525437259149616380828, 2.48200156597610685879991356126, 2.60696791974846211043663761359, 2.61108377119400778804633141053, 2.74263697392464393422584078254, 2.93937626845352026859992847508, 3.18997448682146557318369095360, 3.37390782374923872688219762636, 3.51886649165010669220867172732, 3.54711612946183544983508582897, 3.61934492578378904858909251691, 3.78038841065032231900085266698, 3.92330080889104943902022518475, 4.02547163112643252288104562762, 4.04746385607230922513384260244

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

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