Properties

Label 16-605e8-1.1-c1e8-0-5
Degree $16$
Conductor $1.795\times 10^{22}$
Sign $1$
Analytic cond. $296660.$
Root an. cond. $2.19794$
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  − 5·4-s − 10·5-s − 6·9-s + 14·16-s + 50·20-s + 45·25-s + 40·31-s + 30·36-s + 60·45-s − 10·49-s + 8·59-s − 40·64-s − 16·71-s − 140·80-s + 9·81-s − 225·100-s − 200·124-s − 100·125-s + 127-s + 131-s + 137-s + 139-s − 84·144-s + 149-s + 151-s − 400·155-s + 157-s + ⋯
L(s)  = 1  − 5/2·4-s − 4.47·5-s − 2·9-s + 7/2·16-s + 11.1·20-s + 9·25-s + 7.18·31-s + 5·36-s + 8.94·45-s − 1.42·49-s + 1.04·59-s − 5·64-s − 1.89·71-s − 15.6·80-s + 81-s − 22.5·100-s − 17.9·124-s − 8.94·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7·144-s + 0.0819·149-s + 0.0813·151-s − 32.1·155-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(296660.\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2031092474\)
\(L(\frac12)\) \(\approx\) \(0.2031092474\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \)
11 \( 1 \)
good2 \( 1 + 5 T^{2} + 11 T^{4} + 25 T^{6} + 61 T^{8} + 25 p^{2} T^{10} + 11 p^{4} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} \)
3 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
7 \( 1 + 10 T^{2} + 11 T^{4} + 200 T^{6} + 3781 T^{8} + 200 p^{2} T^{10} + 11 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 50 T^{2} + 1331 T^{4} - 25000 T^{6} + 363061 T^{8} - 25000 p^{2} T^{10} + 1331 p^{4} T^{12} - 50 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 + 70 T^{2} + 2651 T^{4} + 68600 T^{6} + 1327141 T^{8} + 68600 p^{2} T^{10} + 2651 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - p T^{2} )^{8} \)
29 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 20 T + 209 T^{2} - 1600 T^{3} + 9841 T^{4} - 1600 p T^{5} + 209 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 3522 T^{4} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 4 T - 43 T^{2} + 408 T^{3} + 905 T^{4} + 408 p T^{5} - 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - p T^{2} )^{8} \)
71 \( ( 1 + 8 T - 7 T^{2} - 624 T^{3} - 4495 T^{4} - 624 p T^{5} - 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 10 T^{2} - 5269 T^{4} + 200 T^{6} + 28505221 T^{8} + 200 p^{2} T^{10} - 5269 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
83 \( 1 - 370 T^{2} + 75251 T^{4} - 10130600 T^{6} + 985743541 T^{8} - 10130600 p^{2} T^{10} + 75251 p^{4} T^{12} - 370 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
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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

−4.49296070572724849731893375169, −4.44511932053291882400995129062, −4.36051420484588114191276144499, −4.29822570695035416665600697241, −4.06733012926720557251421115441, −4.05651294628167799671206376295, −4.01821926490180502078822441261, −3.81641808073273028376633041448, −3.39088455990360914138046953108, −3.37616625491496910978177356124, −3.24209528170556487090054960087, −3.12475276566318883998019062865, −3.07523112776643168439152179468, −3.00535894091498237184184576529, −2.69139613564784564134322976748, −2.57063137206641762348552611520, −2.43251737404647296561262193985, −2.12455713919972767991096341077, −1.68066974952694402511355172313, −1.52191771568781573467269811470, −1.09324233311305879768541280273, −0.827110009784552620750718314605, −0.74965194608776641336744147747, −0.36568054140630699982563828319, −0.27026797428729992857328065727, 0.27026797428729992857328065727, 0.36568054140630699982563828319, 0.74965194608776641336744147747, 0.827110009784552620750718314605, 1.09324233311305879768541280273, 1.52191771568781573467269811470, 1.68066974952694402511355172313, 2.12455713919972767991096341077, 2.43251737404647296561262193985, 2.57063137206641762348552611520, 2.69139613564784564134322976748, 3.00535894091498237184184576529, 3.07523112776643168439152179468, 3.12475276566318883998019062865, 3.24209528170556487090054960087, 3.37616625491496910978177356124, 3.39088455990360914138046953108, 3.81641808073273028376633041448, 4.01821926490180502078822441261, 4.05651294628167799671206376295, 4.06733012926720557251421115441, 4.29822570695035416665600697241, 4.36051420484588114191276144499, 4.44511932053291882400995129062, 4.49296070572724849731893375169

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

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