↑

Properties

Label 16-85e8-1.1-c1e8-0-1
Degree $16$
Conductor $2.725\times 10^{15}$
Sign $1$
Analytic cond. $0.0450368$
Root an. cond. $0.823849$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s − 2·5-s + 8·9-s − 4·11-s + 2·16-s − 8·19-s − 8·20-s − 4·25-s + 8·29-s − 24·31-s + 32·36-s − 12·41-s − 16·44-s − 16·45-s + 20·49-s + 8·55-s + 16·59-s + 12·61-s − 8·64-s − 20·71-s − 32·76-s + 24·79-s − 4·80-s + 24·81-s − 48·89-s + 16·95-s − 32·99-s + ⋯
L(s)  = 1  + 2·4-s − 0.894·5-s + 8/3·9-s − 1.20·11-s + 1/2·16-s − 1.83·19-s − 1.78·20-s − 4/5·25-s + 1.48·29-s − 4.31·31-s + 16/3·36-s − 1.87·41-s − 2.41·44-s − 2.38·45-s + 20/7·49-s + 1.07·55-s + 2.08·59-s + 1.53·61-s − 64-s − 2.37·71-s − 3.67·76-s + 2.70·79-s − 0.447·80-s + 8/3·81-s − 5.08·89-s + 1.64·95-s − 3.21·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(0.0450368\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 17^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9289435161\)
\(L(\frac12)\) \(\approx\) \(0.9289435161\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 2 T + 8 T^{2} + 6 T^{3} + 38 T^{4} + 6 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17 \( ( 1 + T^{2} )^{4} \)
good2 \( 1 - p^{2} T^{2} + 7 p T^{4} - 5 p^{3} T^{6} + 81 T^{8} - 5 p^{5} T^{10} + 7 p^{5} T^{12} - p^{8} T^{14} + p^{8} T^{16} \)
3 \( 1 - 8 T^{2} + 40 T^{4} - 164 T^{6} + 538 T^{8} - 164 p^{2} T^{10} + 40 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 20 T^{2} + 152 T^{4} - 376 T^{6} - 74 p T^{8} - 376 p^{2} T^{10} + 152 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 2 T + 30 T^{2} + 40 T^{3} + 420 T^{4} + 40 p T^{5} + 30 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 12 T^{2} + 316 T^{4} - 5652 T^{6} + 62646 T^{8} - 5652 p^{2} T^{10} + 316 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 4 T + 44 T^{2} + 148 T^{3} + 1062 T^{4} + 148 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 128 T^{2} + 8120 T^{4} - 325980 T^{6} + 8959290 T^{8} - 325980 p^{2} T^{10} + 8120 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 4 T + 84 T^{2} - 204 T^{3} + 3110 T^{4} - 204 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 12 T + 118 T^{2} + 926 T^{3} + 5468 T^{4} + 926 p T^{5} + 118 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 184 T^{2} + 14844 T^{4} - 737480 T^{6} + 28822886 T^{8} - 737480 p^{2} T^{10} + 14844 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 6 T + 128 T^{2} + 586 T^{3} + 7526 T^{4} + 586 p T^{5} + 128 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 212 T^{2} + 23132 T^{4} - 1654508 T^{6} + 83718774 T^{8} - 1654508 p^{2} T^{10} + 23132 p^{4} T^{12} - 212 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 284 T^{2} + 38668 T^{4} - 3256868 T^{6} + 184549366 T^{8} - 3256868 p^{2} T^{10} + 38668 p^{4} T^{12} - 284 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 152 T^{2} + 14780 T^{4} - 1127656 T^{6} + 65975398 T^{8} - 1127656 p^{2} T^{10} + 14780 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 2 T + p T^{2} )^{8} \)
61 \( ( 1 - 6 T + 152 T^{2} - 762 T^{3} + 12982 T^{4} - 762 p T^{5} + 152 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 396 T^{2} + 74604 T^{4} - 8766036 T^{6} + 702758870 T^{8} - 8766036 p^{2} T^{10} + 74604 p^{4} T^{12} - 396 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 10 T + 266 T^{2} + 1932 T^{3} + 27932 T^{4} + 1932 p T^{5} + 266 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 228 T^{2} + 27476 T^{4} - 2839068 T^{6} + 243837526 T^{8} - 2839068 p^{2} T^{10} + 27476 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 12 T + 258 T^{2} - 2274 T^{3} + 27756 T^{4} - 2274 p T^{5} + 258 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 380 T^{2} + 76796 T^{4} - 10351396 T^{6} + 1000791862 T^{8} - 10351396 p^{2} T^{10} + 76796 p^{4} T^{12} - 380 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 24 T + 492 T^{2} + 6276 T^{3} + 71250 T^{4} + 6276 p T^{5} + 492 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 520 T^{2} + 132860 T^{4} - 21638328 T^{6} + 2473797510 T^{8} - 21638328 p^{2} T^{10} + 132860 p^{4} T^{12} - 520 p^{6} T^{14} + p^{8} T^{16} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.84753743345032280555999676393, −6.77106616517837562992908851303, −6.51587921912228671867944115608, −6.44375385686347397013938838855, −6.32858783174428522241835341773, −5.88732080022908782577306448590, −5.60250454769964844469220937588, −5.59578992718384232273264535218, −5.47425074251320789156602539935, −5.15909887959126639872194083157, −5.01790370209473958983877365535, −4.75578477202113841834157296740, −4.52232654991690591442318789920, −4.19908745218452527558406723933, −4.01454426985522740121796022574, −3.84281891265127316846266739828, −3.83429780120465818778217711126, −3.82968295194503621722075766395, −3.05165080139244125986926659785, −2.81598290500285312517941002531, −2.73871140181279190364178298869, −2.13063556655759572533834427822, −2.04510466587025927639381628107, −1.93659467977269682117792101212, −1.48594716639625440615454634913, 1.48594716639625440615454634913, 1.93659467977269682117792101212, 2.04510466587025927639381628107, 2.13063556655759572533834427822, 2.73871140181279190364178298869, 2.81598290500285312517941002531, 3.05165080139244125986926659785, 3.82968295194503621722075766395, 3.83429780120465818778217711126, 3.84281891265127316846266739828, 4.01454426985522740121796022574, 4.19908745218452527558406723933, 4.52232654991690591442318789920, 4.75578477202113841834157296740, 5.01790370209473958983877365535, 5.15909887959126639872194083157, 5.47425074251320789156602539935, 5.59578992718384232273264535218, 5.60250454769964844469220937588, 5.88732080022908782577306448590, 6.32858783174428522241835341773, 6.44375385686347397013938838855, 6.51587921912228671867944115608, 6.77106616517837562992908851303, 6.84753743345032280555999676393

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

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