Properties

Label 16-42e16-1.1-c2e8-0-7
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $2.84884\times 10^{13}$
Root an. cond. $6.93293$
Motivic weight $2$
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  + 88·25-s + 128·37-s + 176·43-s − 256·67-s + 432·79-s − 80·109-s − 240·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.31e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 3.51·25-s + 3.45·37-s + 4.09·43-s − 3.82·67-s + 5.46·79-s − 0.733·109-s − 1.98·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.76·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.84884\times 10^{13}\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(19.49729141\)
\(L(\frac12)\) \(\approx\) \(19.49729141\)
\(L(2)\) 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 \( 1 \)
good5 \( ( 1 - 44 T^{2} + 1636 T^{4} - 44 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 + 120 T^{2} + 1130 T^{4} + 120 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 656 T^{2} + 164608 T^{4} - 656 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 652 T^{2} + 265380 T^{4} - 652 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 - 1308 T^{2} + 687966 T^{4} - 1308 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 + 464 T^{2} + 327738 T^{4} + 464 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 + 1264 T^{2} + 1437274 T^{4} + 1264 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 - 2684 T^{2} + 3534718 T^{4} - 2684 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 32 T + 2112 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 1796 T^{2} + 4042324 T^{4} - 1796 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 - 44 T + 3790 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
47 \( ( 1 - 3908 T^{2} + 6142 p^{2} T^{4} - 3908 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 + 9780 T^{2} + 39185030 T^{4} + 9780 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 8772 T^{2} + 39472926 T^{4} - 8772 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 4496 T^{2} + 31357408 T^{4} - 4496 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 64 T + 6474 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
71 \( ( 1 + 200 p T^{2} + 93659530 T^{4} + 200 p^{5} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 + 3168 T^{2} + 32088096 T^{4} + 3168 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
79 \( ( 1 - 108 T + 9126 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
83 \( ( 1 - 8292 T^{2} + 106271430 T^{4} - 8292 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 8892 T^{2} + 120258948 T^{4} - 8892 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 23328 T^{2} + 262163040 T^{4} - 23328 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−3.72685858293942722842658807549, −3.70802982086312494749666488900, −3.25488686300907784110236395398, −3.21139001673197219738470625858, −3.12205095908222287725468495837, −3.02799017371576129185794418077, −2.86696107061704200978392136466, −2.85381348810941441669802358074, −2.82714083334498221187949179651, −2.77198388360595304288297162286, −2.30948129896880534662582557623, −2.19000042911596960112195400949, −2.13249547233565339260043253024, −2.12043994210072701256284189007, −1.93701137160323376889082953060, −1.69993967646611466577145878666, −1.55569590505110534112640910152, −1.20751416799708808482702427016, −1.14219833787540929477929886745, −0.901120826635624819666934130976, −0.886461995430451802755628795745, −0.820287135466618553171585669069, −0.53456093072238995077945623037, −0.49805089467718059331439919641, −0.19369506153085022773292221853, 0.19369506153085022773292221853, 0.49805089467718059331439919641, 0.53456093072238995077945623037, 0.820287135466618553171585669069, 0.886461995430451802755628795745, 0.901120826635624819666934130976, 1.14219833787540929477929886745, 1.20751416799708808482702427016, 1.55569590505110534112640910152, 1.69993967646611466577145878666, 1.93701137160323376889082953060, 2.12043994210072701256284189007, 2.13249547233565339260043253024, 2.19000042911596960112195400949, 2.30948129896880534662582557623, 2.77198388360595304288297162286, 2.82714083334498221187949179651, 2.85381348810941441669802358074, 2.86696107061704200978392136466, 3.02799017371576129185794418077, 3.12205095908222287725468495837, 3.21139001673197219738470625858, 3.25488686300907784110236395398, 3.70802982086312494749666488900, 3.72685858293942722842658807549

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

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