Properties

Label 32-320e16-1.1-c2e16-0-0
Degree $32$
Conductor $1.209\times 10^{40}$
Sign $1$
Analytic cond. $1.11624\times 10^{15}$
Root an. cond. $2.95285$
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  + 24·9-s + 16·25-s − 48·41-s − 568·49-s + 192·81-s − 1.15e3·89-s − 176·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.60e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 384·225-s + ⋯
L(s)  = 1  + 8/3·9-s + 0.639·25-s − 1.17·41-s − 11.5·49-s + 2.37·81-s − 12.9·89-s − 1.45·121-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 − 9.46·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 1.70·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{96} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.11624\times 10^{15}\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{96} \cdot 5^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.003563061137\)
\(L(\frac12)\) \(\approx\) \(0.003563061137\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - 8 T^{2} + 6 p^{3} T^{4} - 8 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
good3 \( ( 1 - 2 p T^{2} + 14 p T^{4} - 2 p^{5} T^{6} + p^{8} T^{8} )^{4} \)
7 \( ( 1 + 142 T^{2} + 9714 T^{4} + 142 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
11 \( ( 1 + 4 p T^{2} - 3258 T^{4} + 4 p^{5} T^{6} + p^{8} T^{8} )^{4} \)
13 \( ( 1 + 400 T^{2} + 84222 T^{4} + 400 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
17 \( ( 1 - 604 T^{2} + 206646 T^{4} - 604 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
19 \( ( 1 + 614 T^{2} + p^{4} T^{4} )^{8} \)
23 \( ( 1 + 2 p T^{2} + 275250 T^{4} + 2 p^{5} T^{6} + p^{8} T^{8} )^{4} \)
29 \( ( 1 - 2420 T^{2} + 2672262 T^{4} - 2420 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
31 \( ( 1 - 2804 T^{2} + 3804390 T^{4} - 2804 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
37 \( ( 1 + 3640 T^{2} + 6944622 T^{4} + 3640 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
41 \( ( 1 + 6 T + 2210 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{8} \)
43 \( ( 1 - 4102 T^{2} + 9879978 T^{4} - 4102 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
47 \( ( 1 + 6286 T^{2} + 18424050 T^{4} + 6286 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
53 \( ( 1 + 4912 T^{2} + 13750398 T^{4} + 4912 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
59 \( ( 1 + 3020 T^{2} - 3206778 T^{4} + 3020 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
61 \( ( 1 - 7912 T^{2} + 43299822 T^{4} - 7912 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
67 \( ( 1 - 1270 T^{2} + 38354442 T^{4} - 1270 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
71 \( ( 1 - 19060 T^{2} + 141569958 T^{4} - 19060 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
73 \( ( 1 - 11932 T^{2} + 71334774 T^{4} - 11932 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
79 \( ( 1 + 2188 T^{2} + 44215398 T^{4} + 2188 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
83 \( ( 1 - 8518 T^{2} + 111841962 T^{4} - 8518 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
89 \( ( 1 + 144 T + 20510 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )^{8} \)
97 \( ( 1 - 9820 T^{2} + 155343798 T^{4} - 9820 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.93687169292473492407339804882, −2.91970251105095178939055649214, −2.88386940538432660356771794683, −2.67823553747153115187603253274, −2.67481993328457803333875256532, −2.62833269702359924076322423650, −2.53542953755352751332348217099, −2.33410637529702809213659803348, −2.13135147281065924604166437088, −1.99092005846654008452105236768, −1.84747417330236240268743752510, −1.77404933418191759674797433764, −1.72484140926247125947457229090, −1.64163171073783916632188468547, −1.55984119541742929820027764327, −1.39954541228058453633002029297, −1.34012864614716246248632412525, −1.32177526744949689235902366487, −1.28224537066818358078075610757, −0.996397206492402473603460706553, −0.952288277233522625224967420296, −0.47408398635801451360214302306, −0.35851190103535852873909582706, −0.06092628946667681862994742491, −0.01829118253767435364588827431, 0.01829118253767435364588827431, 0.06092628946667681862994742491, 0.35851190103535852873909582706, 0.47408398635801451360214302306, 0.952288277233522625224967420296, 0.996397206492402473603460706553, 1.28224537066818358078075610757, 1.32177526744949689235902366487, 1.34012864614716246248632412525, 1.39954541228058453633002029297, 1.55984119541742929820027764327, 1.64163171073783916632188468547, 1.72484140926247125947457229090, 1.77404933418191759674797433764, 1.84747417330236240268743752510, 1.99092005846654008452105236768, 2.13135147281065924604166437088, 2.33410637529702809213659803348, 2.53542953755352751332348217099, 2.62833269702359924076322423650, 2.67481993328457803333875256532, 2.67823553747153115187603253274, 2.88386940538432660356771794683, 2.91970251105095178939055649214, 2.93687169292473492407339804882

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

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