Properties

Label 16-1216e8-1.1-c1e8-0-7
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  − 2·5-s − 10·13-s + 10·17-s + 13·25-s − 6·29-s − 24·37-s − 4·41-s − 56·49-s + 10·53-s − 18·61-s + 20·65-s + 28·73-s + 81-s − 20·85-s + 18·89-s − 12·97-s − 22·101-s − 6·109-s − 36·113-s − 14·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 12·145-s + 149-s + ⋯
L(s)  = 1  − 0.894·5-s − 2.77·13-s + 2.42·17-s + 13/5·25-s − 1.11·29-s − 3.94·37-s − 0.624·41-s − 8·49-s + 1.37·53-s − 2.30·61-s + 2.48·65-s + 3.27·73-s + 1/9·81-s − 2.16·85-s + 1.90·89-s − 1.21·97-s − 2.18·101-s − 0.574·109-s − 3.38·113-s − 1.27·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.996·145-s + 0.0819·149-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

\(L(1)\) \(\approx\) \(1.729905465\)
\(L(\frac12)\) \(\approx\) \(1.729905465\)
\(L(\frac{3}{2})\) 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 \( 1 + 35 T^{2} + 36 p T^{4} + 35 p^{2} T^{6} + p^{4} T^{8} \)
good3 \( 1 - T^{4} - 80 T^{8} - p^{4} T^{12} + p^{8} T^{16} \)
5 \( ( 1 + T - p T^{2} - 4 T^{3} + 6 T^{4} - 4 p T^{5} - p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
7 \( ( 1 + p T^{2} )^{8} \)
11 \( ( 1 + 7 T^{2} + 216 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 5 T - 3 T^{2} + 10 T^{3} + 290 T^{4} + 10 p T^{5} - 3 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 5 T - 11 T^{2} - 10 T^{3} + 582 T^{4} - 10 p T^{5} - 11 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 65 T^{2} + 2217 T^{4} - 61750 T^{6} + 1517198 T^{8} - 61750 p^{2} T^{10} + 2217 p^{4} T^{12} - 65 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 3 T - 13 T^{2} - 108 T^{3} - 618 T^{4} - 108 p T^{5} - 13 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 76 T^{2} + 3094 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \)
43 \( 1 - 93 T^{2} + 3133 T^{4} - 169074 T^{6} + 11015454 T^{8} - 169074 p^{2} T^{10} + 3133 p^{4} T^{12} - 93 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 117 T^{2} + 6805 T^{4} - 288522 T^{6} + 12270774 T^{8} - 288522 p^{2} T^{10} + 6805 p^{4} T^{12} - 117 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 5 T - 83 T^{2} - 10 T^{3} + 7530 T^{4} - 10 p T^{5} - 83 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 224 T^{2} + 30687 T^{4} - 2806048 T^{6} + 193041104 T^{8} - 2806048 p^{2} T^{10} + 30687 p^{4} T^{12} - 224 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 + 9 T - 57 T^{2} + 144 T^{3} + 10382 T^{4} + 144 p T^{5} - 57 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 40 T^{2} - 1641 T^{4} + 229480 T^{6} - 16829440 T^{8} + 229480 p^{2} T^{10} - 1641 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 257 T^{2} + 39561 T^{4} - 4216342 T^{6} + 343249454 T^{8} - 4216342 p^{2} T^{10} + 39561 p^{4} T^{12} - 257 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 - 14 T + 69 T^{2} + 266 T^{3} - 3508 T^{4} + 266 p T^{5} + 69 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 - 245 T^{2} + 33493 T^{4} - 3442250 T^{6} + 290380918 T^{8} - 3442250 p^{2} T^{10} + 33493 p^{4} T^{12} - 245 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 291 T^{2} + 34604 T^{4} + 291 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 9 T - 79 T^{2} + 162 T^{3} + 10470 T^{4} + 162 p T^{5} - 79 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 6 T - 99 T^{2} - 354 T^{3} + 5324 T^{4} - 354 p T^{5} - 99 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.13371225680253503070498848818, −4.11166258865235076864890157345, −3.74221834661896417470723075958, −3.73971789699617590188181979707, −3.63457483749011420599465803629, −3.40523515343781593190503374611, −3.29392176849403236236755819825, −3.24329474933190390791436004756, −2.96315952666570897498502079853, −2.95511754868336794445868667760, −2.94281004361226308560632532446, −2.87163677589126435014264446968, −2.86199543812646480683781857797, −2.15639424872154195928723125206, −2.08264371981845004653990417402, −2.03186209115015307330478902128, −2.01251757137446784241719311740, −1.78690457793055622819268910912, −1.49161751235416593924183436488, −1.43799690357337026626292481918, −1.23195746984881035443040608963, −1.06105465768708348292029313173, −0.46435937035163675947438230143, −0.37214293057939787788812773611, −0.29212685209668569000213088047, 0.29212685209668569000213088047, 0.37214293057939787788812773611, 0.46435937035163675947438230143, 1.06105465768708348292029313173, 1.23195746984881035443040608963, 1.43799690357337026626292481918, 1.49161751235416593924183436488, 1.78690457793055622819268910912, 2.01251757137446784241719311740, 2.03186209115015307330478902128, 2.08264371981845004653990417402, 2.15639424872154195928723125206, 2.86199543812646480683781857797, 2.87163677589126435014264446968, 2.94281004361226308560632532446, 2.95511754868336794445868667760, 2.96315952666570897498502079853, 3.24329474933190390791436004756, 3.29392176849403236236755819825, 3.40523515343781593190503374611, 3.63457483749011420599465803629, 3.73971789699617590188181979707, 3.74221834661896417470723075958, 4.11166258865235076864890157345, 4.13371225680253503070498848818

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

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