Properties

Label 16-546e8-1.1-c1e8-0-6
Degree $16$
Conductor $7.898\times 10^{21}$
Sign $1$
Analytic cond. $130544.$
Root an. cond. $2.08802$
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  − 4·5-s − 4·9-s − 8·11-s + 16·13-s − 2·16-s − 12·17-s + 4·19-s + 8·25-s − 12·29-s + 20·31-s − 8·37-s + 16·41-s + 16·45-s − 16·47-s − 2·49-s − 24·53-s + 32·55-s + 28·59-s − 64·65-s + 8·67-s + 52·71-s − 8·73-s − 48·79-s + 8·80-s + 10·81-s + 32·83-s + 48·85-s + ⋯
L(s)  = 1  − 1.78·5-s − 4/3·9-s − 2.41·11-s + 4.43·13-s − 1/2·16-s − 2.91·17-s + 0.917·19-s + 8/5·25-s − 2.22·29-s + 3.59·31-s − 1.31·37-s + 2.49·41-s + 2.38·45-s − 2.33·47-s − 2/7·49-s − 3.29·53-s + 4.31·55-s + 3.64·59-s − 7.93·65-s + 0.977·67-s + 6.17·71-s − 0.936·73-s − 5.40·79-s + 0.894·80-s + 10/9·81-s + 3.51·83-s + 5.20·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(130544.\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7556019574\)
\(L(\frac12)\) \(\approx\) \(0.7556019574\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( ( 1 + T^{2} )^{4} \)
7 \( 1 + 2 T^{2} + 24 T^{3} + 2 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \)
13 \( ( 1 - 4 T + p T^{2} )^{4} \)
good5 \( 1 + 4 T + 8 T^{2} + 16 T^{3} - 7 T^{4} - 96 T^{5} - 8 p^{2} T^{6} - 84 p T^{7} - 864 T^{8} - 84 p^{2} T^{9} - 8 p^{4} T^{10} - 96 p^{3} T^{11} - 7 p^{4} T^{12} + 16 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
11 \( 1 + 8 T + 32 T^{2} + 104 T^{3} + 425 T^{4} + 1984 T^{5} + 7680 T^{6} + 26384 T^{7} + 88768 T^{8} + 26384 p T^{9} + 7680 p^{2} T^{10} + 1984 p^{3} T^{11} + 425 p^{4} T^{12} + 104 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 + 6 T + 33 T^{2} + 6 p T^{3} + 452 T^{4} + 6 p^{2} T^{5} + 33 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 4 T + 8 T^{2} + 108 T^{3} - 923 T^{4} + 3448 T^{5} - 576 T^{6} - 62104 T^{7} + 454164 T^{8} - 62104 p T^{9} - 576 p^{2} T^{10} + 3448 p^{3} T^{11} - 923 p^{4} T^{12} + 108 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 82 T^{2} + 4001 T^{4} - 133098 T^{6} + 150460 p T^{8} - 133098 p^{2} T^{10} + 4001 p^{4} T^{12} - 82 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 6 T + 49 T^{2} + 262 T^{3} + 1516 T^{4} + 262 p T^{5} + 49 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 - 20 T + 200 T^{2} - 1620 T^{3} + 12688 T^{4} - 90100 T^{5} + 600 p^{2} T^{6} - 115340 p T^{7} + 20991006 T^{8} - 115340 p^{2} T^{9} + 600 p^{4} T^{10} - 90100 p^{3} T^{11} + 12688 p^{4} T^{12} - 1620 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 8 T + 32 T^{2} + 124 T^{3} - 1323 T^{4} - 4504 T^{5} + 13992 T^{6} + 419908 T^{7} + 5808148 T^{8} + 419908 p T^{9} + 13992 p^{2} T^{10} - 4504 p^{3} T^{11} - 1323 p^{4} T^{12} + 124 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 16 T + 128 T^{2} - 976 T^{3} + 4868 T^{4} + 3472 T^{5} - 202368 T^{6} + 2524880 T^{7} - 21791162 T^{8} + 2524880 p T^{9} - 202368 p^{2} T^{10} + 3472 p^{3} T^{11} + 4868 p^{4} T^{12} - 976 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
43 \( 1 - 90 T^{2} + 10113 T^{4} - 530762 T^{6} + 30682740 T^{8} - 530762 p^{2} T^{10} + 10113 p^{4} T^{12} - 90 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 16 T + 128 T^{2} + 1072 T^{3} + 8612 T^{4} + 54512 T^{5} + 344448 T^{6} + 2597200 T^{7} + 19498438 T^{8} + 2597200 p T^{9} + 344448 p^{2} T^{10} + 54512 p^{3} T^{11} + 8612 p^{4} T^{12} + 1072 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
53 \( ( 1 + 12 T + 208 T^{2} + 1780 T^{3} + 16558 T^{4} + 1780 p T^{5} + 208 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 28 T + 392 T^{2} - 3724 T^{3} + 35888 T^{4} - 387996 T^{5} + 3729880 T^{6} - 28337484 T^{7} + 206048766 T^{8} - 28337484 p T^{9} + 3729880 p^{2} T^{10} - 387996 p^{3} T^{11} + 35888 p^{4} T^{12} - 3724 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 382 T^{2} + 68881 T^{4} - 7610686 T^{6} + 561621556 T^{8} - 7610686 p^{2} T^{10} + 68881 p^{4} T^{12} - 382 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 8 T + 32 T^{2} + 264 T^{3} + 2124 T^{4} - 71448 T^{5} + 538464 T^{6} - 1186856 T^{7} - 21425914 T^{8} - 1186856 p T^{9} + 538464 p^{2} T^{10} - 71448 p^{3} T^{11} + 2124 p^{4} T^{12} + 264 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 52 T + 1352 T^{2} - 23940 T^{3} + 329488 T^{4} - 3791876 T^{5} + 38271576 T^{6} - 351847604 T^{7} + 3038803806 T^{8} - 351847604 p T^{9} + 38271576 p^{2} T^{10} - 3791876 p^{3} T^{11} + 329488 p^{4} T^{12} - 23940 p^{5} T^{13} + 1352 p^{6} T^{14} - 52 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 8 T + 32 T^{2} + 412 T^{3} + 6885 T^{4} + 48056 T^{5} + 249000 T^{6} + 4429588 T^{7} + 78914788 T^{8} + 4429588 p T^{9} + 249000 p^{2} T^{10} + 48056 p^{3} T^{11} + 6885 p^{4} T^{12} + 412 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
79 \( ( 1 + 24 T + 314 T^{2} + 2208 T^{3} + 16386 T^{4} + 2208 p T^{5} + 314 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 32 T + 512 T^{2} - 7040 T^{3} + 95108 T^{4} - 1094080 T^{5} + 11096064 T^{6} - 113120096 T^{7} + 1100688934 T^{8} - 113120096 p T^{9} + 11096064 p^{2} T^{10} - 1094080 p^{3} T^{11} + 95108 p^{4} T^{12} - 7040 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 4 T + 8 T^{2} + 84 T^{3} - 11488 T^{4} + 30988 T^{5} - 28520 T^{6} - 307740 T^{7} + 57762814 T^{8} - 307740 p T^{9} - 28520 p^{2} T^{10} + 30988 p^{3} T^{11} - 11488 p^{4} T^{12} + 84 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 36 T + 648 T^{2} + 8028 T^{3} + 63664 T^{4} + 84996 T^{5} - 5970024 T^{6} - 114051780 T^{7} - 1348791330 T^{8} - 114051780 p T^{9} - 5970024 p^{2} T^{10} + 84996 p^{3} T^{11} + 63664 p^{4} T^{12} + 8028 p^{5} T^{13} + 648 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.64924460691354417162145725755, −4.63006749680732696696892145215, −4.57055922316804466584229718575, −4.25755149935077928625378617282, −4.02796505324098686893223794570, −3.90453265628068329394744023935, −3.80129379115286667880165489356, −3.78388297919481536906324284479, −3.71350996810764682342609927592, −3.67723885180427102129972312479, −3.29988634223920636584905149557, −3.03765288550293821321450656340, −2.84066302175862214618169806178, −2.79998480808788510229249804935, −2.70111664539984533557250764507, −2.49878100154280320485867008824, −2.45310539946190554623015387883, −2.34419270266775474645925308968, −1.63211104732944702899421863257, −1.63046299330269929145389754242, −1.51333430451822239798204384811, −1.28549466134834050305842340829, −0.885573481062164791498229567993, −0.37874954460619744472812084669, −0.29433393339490897134034674654, 0.29433393339490897134034674654, 0.37874954460619744472812084669, 0.885573481062164791498229567993, 1.28549466134834050305842340829, 1.51333430451822239798204384811, 1.63046299330269929145389754242, 1.63211104732944702899421863257, 2.34419270266775474645925308968, 2.45310539946190554623015387883, 2.49878100154280320485867008824, 2.70111664539984533557250764507, 2.79998480808788510229249804935, 2.84066302175862214618169806178, 3.03765288550293821321450656340, 3.29988634223920636584905149557, 3.67723885180427102129972312479, 3.71350996810764682342609927592, 3.78388297919481536906324284479, 3.80129379115286667880165489356, 3.90453265628068329394744023935, 4.02796505324098686893223794570, 4.25755149935077928625378617282, 4.57055922316804466584229718575, 4.63006749680732696696892145215, 4.64924460691354417162145725755

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

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