Properties

Label 16-72e8-1.1-c2e8-0-0
Degree $16$
Conductor $7.222\times 10^{14}$
Sign $1$
Analytic cond. $219.452$
Root an. cond. $1.40066$
Motivic weight $2$
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  + 2·4-s − 16·16-s − 80·25-s − 128·31-s − 184·49-s − 40·64-s − 160·73-s − 384·79-s − 192·97-s − 160·100-s + 896·103-s − 360·121-s − 256·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 408·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 1/2·4-s − 16-s − 3.19·25-s − 4.12·31-s − 3.75·49-s − 5/8·64-s − 2.19·73-s − 4.86·79-s − 1.97·97-s − 8/5·100-s + 8.69·103-s − 2.97·121-s − 2.06·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.41·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(219.452\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{72} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.008255849380\)
\(L(\frac12)\) \(\approx\) \(0.008255849380\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T^{2} + 5 p^{2} T^{4} - p^{5} T^{6} + p^{8} T^{8} \)
3 \( 1 \)
good5 \( ( 1 + 8 p T^{2} + 818 T^{4} + 8 p^{5} T^{6} + p^{8} T^{8} )^{2} \)
7 \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{4} \)
11 \( ( 1 + 180 T^{2} + 24070 T^{4} + 180 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 204 T^{2} + 14278 T^{4} - 204 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 320 T^{2} + 189314 T^{4} - 320 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 + 60 T^{2} + 48550 T^{4} + 60 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 - 1652 T^{2} + 1188710 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 + 2376 T^{2} + 2685298 T^{4} + 2376 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 + 32 T + 1710 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 3972 T^{2} + 7479526 T^{4} - 3972 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
41 \( ( 1 - 5792 T^{2} + 13955138 T^{4} - 5792 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 - 996 T^{2} + 6872614 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 - 4660 T^{2} + 10875174 T^{4} - 4660 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 + 8712 T^{2} + 34754866 T^{4} + 8712 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 + 9316 T^{2} + 45079718 T^{4} + 9316 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 9412 T^{2} + 45525030 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
67 \( ( 1 - 12484 T^{2} + 74951718 T^{4} - 12484 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 9076 T^{2} + 54164454 T^{4} - 9076 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 + 40 T + 7730 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
79 \( ( 1 + 96 T + 8494 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
83 \( ( 1 + 24436 T^{2} + 241946438 T^{4} + 24436 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 25856 T^{2} + 284297666 T^{4} - 25856 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 48 T + 18562 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−6.91246096114374111982788961004, −6.38124805760637859428391653241, −6.37759977511157285116304916054, −6.23308521978685461767690093848, −5.96115217893513013603737199003, −5.72629301603921812389749145900, −5.68776511745155046980242734231, −5.53972873926725916349702365393, −5.33400493619717620391159653240, −4.99274355783893844568540632475, −4.76502272990261567986439284145, −4.66953098928050755540552697235, −4.34129296725089818607756519613, −4.18697262229911461909533270410, −3.77811778538907013750361112231, −3.72856895031028334834369935121, −3.68495211804995054709706564341, −3.07227801042497684303982487122, −2.92611784155963784839975619419, −2.86089602535425140732862029318, −2.13228612868530461579985154647, −1.91142564045386729776039079357, −1.71811595376210578469323126760, −1.59190263552323676275955853787, −0.02820736993504295107020202610, 0.02820736993504295107020202610, 1.59190263552323676275955853787, 1.71811595376210578469323126760, 1.91142564045386729776039079357, 2.13228612868530461579985154647, 2.86089602535425140732862029318, 2.92611784155963784839975619419, 3.07227801042497684303982487122, 3.68495211804995054709706564341, 3.72856895031028334834369935121, 3.77811778538907013750361112231, 4.18697262229911461909533270410, 4.34129296725089818607756519613, 4.66953098928050755540552697235, 4.76502272990261567986439284145, 4.99274355783893844568540632475, 5.33400493619717620391159653240, 5.53972873926725916349702365393, 5.68776511745155046980242734231, 5.72629301603921812389749145900, 5.96115217893513013603737199003, 6.23308521978685461767690093848, 6.37759977511157285116304916054, 6.38124805760637859428391653241, 6.91246096114374111982788961004

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

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