Properties

Label 16-320e8-1.1-c1e8-0-2
Degree $16$
Conductor $1.100\times 10^{20}$
Sign $1$
Analytic cond. $1817.25$
Root an. cond. $1.59850$
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  − 4·25-s − 32·41-s + 48·49-s − 36·81-s − 16·89-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 4/5·25-s − 4.99·41-s + 48/7·49-s − 4·81-s − 1.69·89-s + 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1817.25\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.350790496\)
\(L(\frac12)\) \(\approx\) \(2.350790496\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
good3 \( ( 1 + p^{2} T^{4} )^{4} \)
7 \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 4 T + p T^{2} )^{8} \)
43 \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 + p T^{2} )^{8} \)
59 \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + 2 T + p T^{2} )^{8} \)
97 \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \)
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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

−5.22897475258815021567475848221, −5.05116399024497411157962406659, −4.96603103366504215898698420302, −4.86721350314232118735051269196, −4.44542507118119795734508654157, −4.40293352949898722162853413851, −4.37341784841127647492458158909, −4.18454433343679117702411069805, −3.90293463602792371090500467811, −3.85017615613023600042233018883, −3.81272608323679950141687032036, −3.41959735746845665314996242588, −3.28437876669398359447693629076, −3.21517462654926945323503370272, −2.97706620355683093438152571263, −2.80572461435189366092514981619, −2.52952004466863090983915260680, −2.40054514738793687255700269314, −2.19495323186898686631424848099, −1.91140774477372873124728056901, −1.74640248866218297015733100107, −1.46150173448109023417904371919, −1.33870051875073351926471888572, −0.65906081523396271103566142236, −0.48944471103608752263182176230, 0.48944471103608752263182176230, 0.65906081523396271103566142236, 1.33870051875073351926471888572, 1.46150173448109023417904371919, 1.74640248866218297015733100107, 1.91140774477372873124728056901, 2.19495323186898686631424848099, 2.40054514738793687255700269314, 2.52952004466863090983915260680, 2.80572461435189366092514981619, 2.97706620355683093438152571263, 3.21517462654926945323503370272, 3.28437876669398359447693629076, 3.41959735746845665314996242588, 3.81272608323679950141687032036, 3.85017615613023600042233018883, 3.90293463602792371090500467811, 4.18454433343679117702411069805, 4.37341784841127647492458158909, 4.40293352949898722162853413851, 4.44542507118119795734508654157, 4.86721350314232118735051269196, 4.96603103366504215898698420302, 5.05116399024497411157962406659, 5.22897475258815021567475848221

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

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