Properties

Label 16-300e8-1.1-c1e8-0-6
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $1084.39$
Root an. cond. $1.54774$
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-s + 9-s − 4·13-s + 3·16-s − 36-s − 8·37-s + 18·49-s + 4·52-s − 20·61-s − 9·64-s + 36·73-s − 7·81-s + 12·97-s + 44·109-s − 4·117-s − 30·121-s + 127-s + 131-s + 137-s + 139-s + 3·144-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/2·4-s + 1/3·9-s − 1.10·13-s + 3/4·16-s − 1/6·36-s − 1.31·37-s + 18/7·49-s + 0.554·52-s − 2.56·61-s − 9/8·64-s + 4.21·73-s − 7/9·81-s + 1.21·97-s + 4.21·109-s − 0.369·117-s − 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/4·144-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1084.39\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.537305868\)
\(L(\frac12)\) \(\approx\) \(1.537305868\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{2} - p T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
3 \( 1 - T^{2} + 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
5 \( 1 \)
good7 \( ( 1 - 9 T^{2} + 108 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 15 T^{2} + 288 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 9 T^{2} - 232 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 50 T^{2} + 1183 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 64 T^{2} + 1918 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 80 T^{2} + 3118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 69 T^{2} + 2856 T^{4} - 69 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 49 T^{2} + 1656 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 101 T^{2} + 5008 T^{4} - 101 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 136 T^{2} + 8878 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 176 T^{2} + 13198 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + p T^{2} )^{8} \)
61 \( ( 1 + 5 T + 118 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 234 T^{2} + 22503 T^{4} - 234 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 9918 T^{4} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 9 T + 156 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 240 T^{2} + 26718 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 319 T^{2} + 39208 T^{4} + 319 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 305 T^{2} + 39088 T^{4} - 305 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−5.27250910433095717077873714338, −5.10488937975200745086128154332, −4.89752674401541881697235875089, −4.73677071918912998673428356577, −4.71501548691725087858585452637, −4.70914271532875139571447180037, −4.47181657406146890817701410400, −4.12265426821923548189534241450, −3.93552589258612856189814586927, −3.85252059322530232028444401677, −3.72977384243359151588540675288, −3.71669284931644949903297781325, −3.40876520709646186347334872651, −3.18540865369009122824476435331, −2.92526794406119207142279040446, −2.85796468558204773986180809540, −2.71306679449250231284640212447, −2.39910027569157357029289583304, −2.16702125005660435668153660174, −1.99826884252279656142511927702, −1.79824033868305525880523082077, −1.56862960486840238227799473619, −1.08287917129573480032568601736, −0.900346535699855662153296117377, −0.37040779619600545298043630020, 0.37040779619600545298043630020, 0.900346535699855662153296117377, 1.08287917129573480032568601736, 1.56862960486840238227799473619, 1.79824033868305525880523082077, 1.99826884252279656142511927702, 2.16702125005660435668153660174, 2.39910027569157357029289583304, 2.71306679449250231284640212447, 2.85796468558204773986180809540, 2.92526794406119207142279040446, 3.18540865369009122824476435331, 3.40876520709646186347334872651, 3.71669284931644949903297781325, 3.72977384243359151588540675288, 3.85252059322530232028444401677, 3.93552589258612856189814586927, 4.12265426821923548189534241450, 4.47181657406146890817701410400, 4.70914271532875139571447180037, 4.71501548691725087858585452637, 4.73677071918912998673428356577, 4.89752674401541881697235875089, 5.10488937975200745086128154332, 5.27250910433095717077873714338

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

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