Properties

Label 16-1216e8-1.1-c1e8-0-2
Degree $16$
Conductor $4.780\times 10^{24}$
Sign $1$
Analytic cond. $7.90106\times 10^{7}$
Root an. cond. $3.11605$
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  − 4·9-s − 24·17-s − 8·25-s + 52·49-s + 56·73-s − 26·81-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 96·153-s + 157-s + 163-s + 167-s − 76·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 4/3·9-s − 5.82·17-s − 8/5·25-s + 52/7·49-s + 6.55·73-s − 2.88·81-s − 8·121-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 + 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 5.84·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(7.90106\times 10^{7}\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5045398401\)
\(L(\frac12)\) \(\approx\) \(0.5045398401\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
good3 \( ( 1 + T^{2} + p^{2} T^{4} )^{4} \)
5 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \)
7 \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 + p T^{2} )^{8} \)
13 \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 + 3 T + p T^{2} )^{8} \)
23 \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + p T^{2} )^{8} \)
37 \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 - p T^{2} )^{8} \)
53 \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \)
67 \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 7 T + p T^{2} )^{8} \)
79 \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \)
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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

−4.14718359850822826139172973262, −4.06221029139059275431661223399, −4.01017026274360063816103681163, −3.96724887507630970068491393738, −3.67219907490891301998952603979, −3.58103567637207080362136103586, −3.36092327294696662444041609657, −3.30990611421250955451208655633, −2.96014391000346743989950961517, −2.92841693220948713389994166421, −2.58731939694500820612134034625, −2.51939043999563612524244591146, −2.45215125955196814113054812171, −2.36995107046449155993594136316, −2.31111772095551403053240269003, −2.30826587226868188248152410530, −1.98163288042046773655853772138, −1.91414723239338333977799859290, −1.56894531635064829727391448516, −1.42459055477061430825202347179, −1.17975443599525605456101406317, −0.857626994542693978182521476933, −0.61462210392373565737251189020, −0.41008454955637129529280685348, −0.12294204349789222863416519535, 0.12294204349789222863416519535, 0.41008454955637129529280685348, 0.61462210392373565737251189020, 0.857626994542693978182521476933, 1.17975443599525605456101406317, 1.42459055477061430825202347179, 1.56894531635064829727391448516, 1.91414723239338333977799859290, 1.98163288042046773655853772138, 2.30826587226868188248152410530, 2.31111772095551403053240269003, 2.36995107046449155993594136316, 2.45215125955196814113054812171, 2.51939043999563612524244591146, 2.58731939694500820612134034625, 2.92841693220948713389994166421, 2.96014391000346743989950961517, 3.30990611421250955451208655633, 3.36092327294696662444041609657, 3.58103567637207080362136103586, 3.67219907490891301998952603979, 3.96724887507630970068491393738, 4.01017026274360063816103681163, 4.06221029139059275431661223399, 4.14718359850822826139172973262

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

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