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Properties

Label 16-630e8-1.1-c1e8-0-6
Degree $16$
Conductor $2.482\times 10^{22}$
Sign $1$
Analytic cond. $410148.$
Root an. cond. $2.24289$
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  − 4·13-s − 2·16-s + 8·23-s − 2·25-s − 24·29-s − 8·31-s − 4·37-s − 12·43-s − 12·47-s + 32·53-s − 16·59-s + 20·67-s + 36·73-s + 56·83-s − 72·89-s − 4·97-s − 24·103-s + 16·107-s + 32·113-s + 12·121-s − 16·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.10·13-s − 1/2·16-s + 1.66·23-s − 2/5·25-s − 4.45·29-s − 1.43·31-s − 0.657·37-s − 1.82·43-s − 1.75·47-s + 4.39·53-s − 2.08·59-s + 2.44·67-s + 4.21·73-s + 6.14·83-s − 7.63·89-s − 0.406·97-s − 2.36·103-s + 1.54·107-s + 3.01·113-s + 1.09·121-s − 1.43·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(410148.\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.902206015\)
\(L(\frac12)\) \(\approx\) \(1.902206015\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + 2 T^{2} + 16 T^{3} + 2 T^{4} + 16 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( 1 - 12 T^{2} + 40 T^{4} - 260 T^{6} + 15086 T^{8} - 260 p^{2} T^{10} + 40 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 + 4 T + 8 T^{2} + 28 T^{3} + 80 T^{4} + 372 T^{5} + 1240 T^{6} + 6412 T^{7} + 34174 T^{8} + 6412 p T^{9} + 1240 p^{2} T^{10} + 372 p^{3} T^{11} + 80 p^{4} T^{12} + 28 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 64 T^{3} - 252 T^{4} - 960 T^{5} + 2048 T^{6} - 6016 T^{7} - 17786 T^{8} - 6016 p T^{9} + 2048 p^{2} T^{10} - 960 p^{3} T^{11} - 252 p^{4} T^{12} + 64 p^{5} T^{13} + p^{8} T^{16} \)
19 \( ( 1 - p T^{2} )^{8} \)
23 \( 1 - 8 T + 32 T^{2} + 8 T^{3} - 924 T^{4} + 5032 T^{5} - 10656 T^{6} - 28456 T^{7} + 420614 T^{8} - 28456 p T^{9} - 10656 p^{2} T^{10} + 5032 p^{3} T^{11} - 924 p^{4} T^{12} + 8 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 + 12 T + 78 T^{2} + 252 T^{3} + 738 T^{4} + 252 p T^{5} + 78 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 4 T + 110 T^{2} + 356 T^{3} + 4930 T^{4} + 356 p T^{5} + 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 4 T + 8 T^{2} + 220 T^{3} - 48 T^{4} - 8684 T^{5} - 10152 T^{6} - 208404 T^{7} - 3196994 T^{8} - 208404 p T^{9} - 10152 p^{2} T^{10} - 8684 p^{3} T^{11} - 48 p^{4} T^{12} + 220 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 76 T^{2} + 5480 T^{4} - 315940 T^{6} + 12648846 T^{8} - 315940 p^{2} T^{10} + 5480 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 12 T + 72 T^{2} + 300 T^{3} - 1008 T^{4} - 17988 T^{5} - 98280 T^{6} - 509316 T^{7} - 2377282 T^{8} - 509316 p T^{9} - 98280 p^{2} T^{10} - 17988 p^{3} T^{11} - 1008 p^{4} T^{12} + 300 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 + 12 T + 72 T^{2} + 220 T^{3} - 2352 T^{4} - 10116 T^{5} + 72152 T^{6} + 1599596 T^{7} + 18271774 T^{8} + 1599596 p T^{9} + 72152 p^{2} T^{10} - 10116 p^{3} T^{11} - 2352 p^{4} T^{12} + 220 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 32 T + 512 T^{2} - 5792 T^{3} + 60388 T^{4} - 619872 T^{5} + 5690880 T^{6} - 44817632 T^{7} + 328258854 T^{8} - 44817632 p T^{9} + 5690880 p^{2} T^{10} - 619872 p^{3} T^{11} + 60388 p^{4} T^{12} - 5792 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 + 8 T + 148 T^{2} + 520 T^{3} + 8710 T^{4} + 520 p T^{5} + 148 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 226 T^{2} + 16 T^{3} + 20162 T^{4} + 16 p T^{5} + 226 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( 1 - 20 T + 200 T^{2} - 1332 T^{3} + 9104 T^{4} - 102596 T^{5} + 1118232 T^{6} - 8938244 T^{7} + 69729150 T^{8} - 8938244 p T^{9} + 1118232 p^{2} T^{10} - 102596 p^{3} T^{11} + 9104 p^{4} T^{12} - 1332 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 216 T^{2} + 31388 T^{4} - 3350120 T^{6} + 263319558 T^{8} - 3350120 p^{2} T^{10} + 31388 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 36 T + 648 T^{2} - 8620 T^{3} + 98192 T^{4} - 980116 T^{5} + 8807960 T^{6} - 73999132 T^{7} + 618591198 T^{8} - 73999132 p T^{9} + 8807960 p^{2} T^{10} - 980116 p^{3} T^{11} + 98192 p^{4} T^{12} - 8620 p^{5} T^{13} + 648 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 280 T^{2} + 32732 T^{4} - 1914792 T^{6} + 96372294 T^{8} - 1914792 p^{2} T^{10} + 32732 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 56 T + 1568 T^{2} - 30056 T^{3} + 449924 T^{4} - 5649064 T^{5} + 62548320 T^{6} - 632947128 T^{7} + 5962291750 T^{8} - 632947128 p T^{9} + 62548320 p^{2} T^{10} - 5649064 p^{3} T^{11} + 449924 p^{4} T^{12} - 30056 p^{5} T^{13} + 1568 p^{6} T^{14} - 56 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 36 T + 734 T^{2} + 10004 T^{3} + 106562 T^{4} + 10004 p T^{5} + 734 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 4 T + 8 T^{2} + 1852 T^{3} + 7856 T^{4} - 48156 T^{5} + 1459480 T^{6} + 10407580 T^{7} - 90680354 T^{8} + 10407580 p T^{9} + 1459480 p^{2} T^{10} - 48156 p^{3} T^{11} + 7856 p^{4} T^{12} + 1852 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.53989150674039207607873383671, −4.52867362730415129420484539348, −4.43970966022175225717561703032, −4.16136773777423730449853580186, −4.01138876178610511851734409188, −3.96165975177953110787876568683, −3.66539121848082707423268610552, −3.64596159764911765178853094983, −3.52200809705384642020716782197, −3.40098260984895911252632608663, −3.21787415023205365356389318800, −3.10239416919458910532689596488, −2.96024461443265103933532551798, −2.67621683874312722685782356352, −2.37185519362567684464101658644, −2.28968812633019286593187512062, −2.27939993445369886582241047100, −1.98008798669324644269097612238, −1.82587699350995261144045061136, −1.71614272522067487779236418573, −1.58650892883983594001900882867, −1.23643976880131727076912142236, −0.75120639775335456668397633283, −0.55125469474807721398218536233, −0.28325936583748507986899964519, 0.28325936583748507986899964519, 0.55125469474807721398218536233, 0.75120639775335456668397633283, 1.23643976880131727076912142236, 1.58650892883983594001900882867, 1.71614272522067487779236418573, 1.82587699350995261144045061136, 1.98008798669324644269097612238, 2.27939993445369886582241047100, 2.28968812633019286593187512062, 2.37185519362567684464101658644, 2.67621683874312722685782356352, 2.96024461443265103933532551798, 3.10239416919458910532689596488, 3.21787415023205365356389318800, 3.40098260984895911252632608663, 3.52200809705384642020716782197, 3.64596159764911765178853094983, 3.66539121848082707423268610552, 3.96165975177953110787876568683, 4.01138876178610511851734409188, 4.16136773777423730449853580186, 4.43970966022175225717561703032, 4.52867362730415129420484539348, 4.53989150674039207607873383671

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

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