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Properties

Label 16-350e8-1.1-c2e8-0-3
Degree $16$
Conductor $2.252\times 10^{20}$
Sign $1$
Analytic cond. $6.84265\times 10^{7}$
Root an. cond. $3.08817$
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  + 4·4-s + 18·9-s + 36·11-s + 4·16-s − 12·19-s − 96·29-s − 84·31-s + 72·36-s + 144·44-s + 20·49-s + 156·59-s − 84·61-s − 16·64-s − 48·71-s − 48·76-s − 220·79-s + 171·81-s + 756·89-s + 648·99-s − 108·101-s − 140·109-s − 384·116-s + 934·121-s − 336·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 4-s + 2·9-s + 3.27·11-s + 1/4·16-s − 0.631·19-s − 3.31·29-s − 2.70·31-s + 2·36-s + 3.27·44-s + 0.408·49-s + 2.64·59-s − 1.37·61-s − 1/4·64-s − 0.676·71-s − 0.631·76-s − 2.78·79-s + 19/9·81-s + 8.49·89-s + 6.54·99-s − 1.06·101-s − 1.28·109-s − 3.31·116-s + 7.71·121-s − 2.70·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.84265\times 10^{7}\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3095570992\)
\(L(\frac12)\) \(\approx\) \(0.3095570992\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
5 \( 1 \)
7 \( 1 - 20 T^{2} + 6 p^{2} T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \)
good3 \( 1 - 2 p^{2} T^{2} + 17 p^{2} T^{4} - 2 p^{4} T^{6} - 44 p^{4} T^{8} - 2 p^{8} T^{10} + 17 p^{10} T^{12} - 2 p^{14} T^{14} + p^{16} T^{16} \)
11 \( ( 1 - 18 T + 19 T^{2} - 1134 T^{3} + 39180 T^{4} - 1134 p^{2} T^{5} + 19 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 412 T^{2} + 89190 T^{4} + 412 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( 1 - 958 T^{2} + 528481 T^{4} - 736702 p^{2} T^{6} + 813124 p^{4} T^{8} - 736702 p^{6} T^{10} + 528481 p^{8} T^{12} - 958 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 + 6 T + 731 T^{2} + 4314 T^{3} + 390972 T^{4} + 4314 p^{2} T^{5} + 731 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 + 1342 T^{2} + 936841 T^{4} + 408559822 T^{6} + 164414007124 T^{8} + 408559822 p^{4} T^{10} + 936841 p^{8} T^{12} + 1342 p^{12} T^{14} + p^{16} T^{16} \)
29 \( ( 1 + 24 T + 1754 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
31 \( ( 1 + 42 T + 1307 T^{2} + 30198 T^{3} + 158508 T^{4} + 30198 p^{2} T^{5} + 1307 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( 1 + 1250 T^{2} + 1851841 T^{4} - 5047078750 T^{6} - 6357137802140 T^{8} - 5047078750 p^{4} T^{10} + 1851841 p^{8} T^{12} + 1250 p^{12} T^{14} + p^{16} T^{16} \)
41 \( ( 1 - 5500 T^{2} + 13185222 T^{4} - 5500 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 - 7244 T^{2} + 19955334 T^{4} - 7244 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( 1 - 3778 T^{2} + 1006153 T^{4} - 13252351282 T^{6} + 82556405566996 T^{8} - 13252351282 p^{4} T^{10} + 1006153 p^{8} T^{12} - 3778 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 + 7618 T^{2} + 556541 p T^{4} + 97177409602 T^{6} + 294199429548196 T^{8} + 97177409602 p^{4} T^{10} + 556541 p^{9} T^{12} + 7618 p^{12} T^{14} + p^{16} T^{16} \)
59 \( ( 1 - 78 T + 5747 T^{2} - 290082 T^{3} + 8773068 T^{4} - 290082 p^{2} T^{5} + 5747 p^{4} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
61 \( ( 1 + 42 T + 2033 T^{2} + 60690 T^{3} - 9569868 T^{4} + 60690 p^{2} T^{5} + 2033 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 + 122 p T^{2} + 23432665 T^{4} + 375683018 p T^{6} + 129513935195284 T^{8} + 375683018 p^{5} T^{10} + 23432665 p^{8} T^{12} + 122 p^{13} T^{14} + p^{16} T^{16} \)
71 \( ( 1 + 12 T + 8318 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
73 \( 1 - 1390 T^{2} - 3571919 T^{4} + 71296523570 T^{6} - 831034108177820 T^{8} + 71296523570 p^{4} T^{10} - 3571919 p^{8} T^{12} - 1390 p^{12} T^{14} + p^{16} T^{16} \)
79 \( ( 1 + 110 T - 2957 T^{2} + 283250 T^{3} + 112247068 T^{4} + 283250 p^{2} T^{5} - 2957 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - 380 T^{2} + 89625894 T^{4} - 380 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 378 T + 71921 T^{2} - 9182754 T^{3} + 904668996 T^{4} - 9182754 p^{2} T^{5} + 71921 p^{4} T^{6} - 378 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 26620 T^{2} + 330657414 T^{4} + 26620 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−4.97063117932728311880090463562, −4.59737095124910731651182159925, −4.50559119289496819163671925701, −4.32324009006454798552347896956, −4.14970898437411901062080958918, −4.11315673466068997260295123285, −3.89592591665607335547653924097, −3.76859034836191955535439119905, −3.68602006274703917896851142318, −3.58077388296765750031845298279, −3.45557167982349398657354372447, −3.21091147338346473225153187818, −3.13994827685516196717009717283, −2.72503627581737195959093490343, −2.30807674049925710909305415363, −2.28820112531440924315362471350, −2.18967846185389159997982615713, −1.95039519993410911285418188148, −1.86615657410001941036490504594, −1.49110801797791140218514207905, −1.37015362635699362355074146019, −1.29420689000514094934392101384, −1.01227755775309271203189754409, −0.62805907411968647615959529145, −0.04315617580185702000233710722, 0.04315617580185702000233710722, 0.62805907411968647615959529145, 1.01227755775309271203189754409, 1.29420689000514094934392101384, 1.37015362635699362355074146019, 1.49110801797791140218514207905, 1.86615657410001941036490504594, 1.95039519993410911285418188148, 2.18967846185389159997982615713, 2.28820112531440924315362471350, 2.30807674049925710909305415363, 2.72503627581737195959093490343, 3.13994827685516196717009717283, 3.21091147338346473225153187818, 3.45557167982349398657354372447, 3.58077388296765750031845298279, 3.68602006274703917896851142318, 3.76859034836191955535439119905, 3.89592591665607335547653924097, 4.11315673466068997260295123285, 4.14970898437411901062080958918, 4.32324009006454798552347896956, 4.50559119289496819163671925701, 4.59737095124910731651182159925, 4.97063117932728311880090463562

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

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