Properties

Label 16-84e16-1.1-c1e8-0-6
Degree $16$
Conductor $6.144\times 10^{30}$
Sign $1$
Analytic cond. $1.01551\times 10^{14}$
Root an. cond. $7.50616$
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  + 8·25-s − 16·29-s − 32·31-s + 16·47-s − 16·53-s − 48·59-s + 16·109-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 8/5·25-s − 2.97·29-s − 5.74·31-s + 2.33·47-s − 2.19·53-s − 6.24·59-s + 1.53·109-s + 4.36·121-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 + 4.92·169-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

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

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.01551\times 10^{14}\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.379982068\)
\(L(\frac12)\) \(\approx\) \(1.379982068\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 8 T^{2} + 24 T^{4} - 8 p T^{6} + 146 T^{8} - 8 p^{3} T^{10} + 24 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 48 T^{2} + 1140 T^{4} - 18576 T^{6} + 231302 T^{8} - 18576 p^{2} T^{10} + 1140 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 32 T^{2} + 592 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( 1 - 72 T^{2} + 2424 T^{4} - 52008 T^{6} + 914354 T^{8} - 52008 p^{2} T^{10} + 2424 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 36 T^{2} + 96 T^{3} + 590 T^{4} + 96 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 - 96 T^{2} + 4980 T^{4} - 175392 T^{6} + 4613990 T^{8} - 175392 p^{2} T^{10} + 4980 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 8 T + 112 T^{2} + 664 T^{3} + 4842 T^{4} + 664 p T^{5} + 112 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 16 T + 180 T^{2} + 1328 T^{3} + 8414 T^{4} + 1328 p T^{5} + 180 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 48 T^{2} + 192 T^{3} + 1778 T^{4} + 192 p T^{5} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( 1 - 248 T^{2} + 29208 T^{4} - 2129944 T^{6} + 2560930 p T^{8} - 2129944 p^{2} T^{10} + 29208 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 24 T^{2} - 324 T^{4} + 63336 T^{6} + 5311334 T^{8} + 63336 p^{2} T^{10} - 324 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 - 8 T + 172 T^{2} - 904 T^{3} + 11358 T^{4} - 904 p T^{5} + 172 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 8 T + 124 T^{2} + 1240 T^{3} + 8310 T^{4} + 1240 p T^{5} + 124 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 24 T + 316 T^{2} + 2808 T^{3} + 21870 T^{4} + 2808 p T^{5} + 316 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 - 192 T^{2} + 19488 T^{4} - 1664448 T^{6} + 117511586 T^{8} - 1664448 p^{2} T^{10} + 19488 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 280 T^{2} + 40956 T^{4} - 4148456 T^{6} + 317117990 T^{8} - 4148456 p^{2} T^{10} + 40956 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 208 T^{2} + 27604 T^{4} - 2815216 T^{6} + 229790374 T^{8} - 2815216 p^{2} T^{10} + 27604 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 256 T^{2} + 38784 T^{4} - 4262912 T^{6} + 356271938 T^{8} - 4262912 p^{2} T^{10} + 38784 p^{4} T^{12} - 256 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 328 T^{2} + 55068 T^{4} - 6274040 T^{6} + 549903302 T^{8} - 6274040 p^{2} T^{10} + 55068 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 76 T^{2} + 192 T^{3} + 11094 T^{4} + 192 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( 1 + 120 T^{2} + 34680 T^{4} + 2809368 T^{6} + 422712050 T^{8} + 2809368 p^{2} T^{10} + 34680 p^{4} T^{12} + 120 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 192 T^{2} + 44928 T^{4} - 5229504 T^{6} + 668134658 T^{8} - 5229504 p^{2} T^{10} + 44928 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.27253584553188334345902422322, −3.16242017882259825489799057814, −3.06512201044111679175307143512, −3.00080467488892753469436342848, −2.89976499777154107160782332529, −2.72223944374069305320843997998, −2.49836074672839114483913709126, −2.48956122292471012835039954641, −2.22892581501558591715071492479, −2.16885156746215684565884441397, −1.94879015211767594670219757309, −1.88242460281998561249784927129, −1.87496522483734251925045749437, −1.74764038244705617405733688838, −1.68902779472810783489439145444, −1.64595592179805648204813733988, −1.61835957062790827618204699634, −1.13434277924406988275178192561, −1.09060526807127929702629432195, −0.966840967612186044873720634701, −0.76585651292532547125401732258, −0.70359030827357003728864488301, −0.23691610720906243972611928420, −0.21690488144836480137114020263, −0.18805878077539160651344737733, 0.18805878077539160651344737733, 0.21690488144836480137114020263, 0.23691610720906243972611928420, 0.70359030827357003728864488301, 0.76585651292532547125401732258, 0.966840967612186044873720634701, 1.09060526807127929702629432195, 1.13434277924406988275178192561, 1.61835957062790827618204699634, 1.64595592179805648204813733988, 1.68902779472810783489439145444, 1.74764038244705617405733688838, 1.87496522483734251925045749437, 1.88242460281998561249784927129, 1.94879015211767594670219757309, 2.16885156746215684565884441397, 2.22892581501558591715071492479, 2.48956122292471012835039954641, 2.49836074672839114483913709126, 2.72223944374069305320843997998, 2.89976499777154107160782332529, 3.00080467488892753469436342848, 3.06512201044111679175307143512, 3.16242017882259825489799057814, 3.27253584553188334345902422322

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

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