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Properties

Label 16-9522e8-1.1-c1e8-0-2
Degree $16$
Conductor $6.758\times 10^{31}$
Sign $1$
Analytic cond. $1.11697\times 10^{15}$
Root an. cond. $8.71972$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 36·4-s − 120·8-s + 12·13-s + 330·16-s − 16·25-s − 96·26-s + 12·29-s − 12·31-s − 792·32-s − 24·41-s − 48·47-s − 36·49-s + 128·50-s + 432·52-s − 96·58-s − 36·59-s + 96·62-s + 1.71e3·64-s − 24·71-s + 12·73-s + 192·82-s + 384·94-s + 288·98-s − 576·100-s − 24·101-s − 1.44e3·104-s + ⋯
L(s)  = 1  − 5.65·2-s + 18·4-s − 42.4·8-s + 3.32·13-s + 82.5·16-s − 3.19·25-s − 18.8·26-s + 2.22·29-s − 2.15·31-s − 140.·32-s − 3.74·41-s − 7.00·47-s − 5.14·49-s + 18.1·50-s + 59.9·52-s − 12.6·58-s − 4.68·59-s + 12.1·62-s + 214.5·64-s − 2.84·71-s + 1.40·73-s + 21.2·82-s + 39.6·94-s + 29.0·98-s − 57.5·100-s − 2.38·101-s − 141.·104-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 23^{16}\)
Sign: $1$
Analytic conductor: \(1.11697\times 10^{15}\)
Root analytic conductor: \(8.71972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{8} \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + 16 T^{2} + 29 p T^{4} + 976 T^{6} + 5296 T^{8} + 976 p^{2} T^{10} + 29 p^{5} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 + 36 T^{2} + 661 T^{4} + 7800 T^{6} + 64572 T^{8} + 7800 p^{2} T^{10} + 661 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 + 28 T^{2} + 589 T^{4} + 8080 T^{6} + 101212 T^{8} + 8080 p^{2} T^{10} + 589 p^{4} T^{12} + 28 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 6 T + 49 T^{2} - 198 T^{3} + 924 T^{4} - 198 p T^{5} + 49 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 + 52 T^{2} + 1192 T^{4} + 860 p T^{6} + 154318 T^{8} + 860 p^{3} T^{10} + 1192 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 + 72 T^{2} + 3052 T^{4} + 87288 T^{6} + 1905990 T^{8} + 87288 p^{2} T^{10} + 3052 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 6 T + 110 T^{2} - 480 T^{3} + 4731 T^{4} - 480 p T^{5} + 110 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 6 T + 73 T^{2} + 18 p T^{3} + 2748 T^{4} + 18 p^{2} T^{5} + 73 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 36 T^{2} + 3064 T^{4} + 129804 T^{6} + 5211534 T^{8} + 129804 p^{2} T^{10} + 3064 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 12 T + 197 T^{2} + 1488 T^{3} + 12780 T^{4} + 1488 p T^{5} + 197 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 + 72 T^{2} + 3628 T^{4} + 195192 T^{6} + 9964038 T^{8} + 195192 p^{2} T^{10} + 3628 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 24 T + 272 T^{2} + 1944 T^{3} + 12318 T^{4} + 1944 p T^{5} + 272 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 + 40 T^{2} + 3574 T^{4} + 111040 T^{6} + 15730795 T^{8} + 111040 p^{2} T^{10} + 3574 p^{4} T^{12} + 40 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 18 T + 341 T^{2} + 3402 T^{3} + 33372 T^{4} + 3402 p T^{5} + 341 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( 1 + 228 T^{2} + 32197 T^{4} + 3033840 T^{6} + 215378124 T^{8} + 3033840 p^{2} T^{10} + 32197 p^{4} T^{12} + 228 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 + 360 T^{2} + 61996 T^{4} + 6779352 T^{6} + 528575814 T^{8} + 6779352 p^{2} T^{10} + 61996 p^{4} T^{12} + 360 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 12 T + 212 T^{2} + 1980 T^{3} + 19254 T^{4} + 1980 p T^{5} + 212 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 6 T + 82 T^{2} - 480 T^{3} + 10095 T^{4} - 480 p T^{5} + 82 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 + 276 T^{2} + 50437 T^{4} + 6055464 T^{6} + 561813948 T^{8} + 6055464 p^{2} T^{10} + 50437 p^{4} T^{12} + 276 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 + 268 T^{2} + 22813 T^{4} - 570560 T^{6} - 191709284 T^{8} - 570560 p^{2} T^{10} + 22813 p^{4} T^{12} + 268 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 + 148 T^{2} + 33637 T^{4} + 3187840 T^{6} + 401195404 T^{8} + 3187840 p^{2} T^{10} + 33637 p^{4} T^{12} + 148 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 + 336 T^{2} + 70558 T^{4} + 10579200 T^{6} + 1162191459 T^{8} + 10579200 p^{2} T^{10} + 70558 p^{4} T^{12} + 336 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.52179843108446757839385189578, −3.16508997064611439462640211727, −3.13782414757194344407617160456, −3.07513658022441424511644034182, −3.02010702653812905332638712944, −2.94517064562137940127330796954, −2.88990372677529900510762076423, −2.85970411739494850604020985321, −2.65561043657055664191972469328, −2.22550419246244089197561048668, −2.09081243277305316383276992632, −2.07532895581553336638670574120, −2.00153486797686611174814276530, −1.98986413176447260101797841218, −1.97401819198881136210101145532, −1.97332002234449329237824128486, −1.59360621940236050959034383426, −1.55295461496433297425722793148, −1.31873950076176200310164566330, −1.28264157129613450745864763285, −1.26503755453763160309475783615, −1.10533774173103271412725863539, −1.10162010241977014115001274333, −1.06287565116037406026740719785, −1.02016128005548230027881818353, 0, 0, 0, 0, 0, 0, 0, 0, 1.02016128005548230027881818353, 1.06287565116037406026740719785, 1.10162010241977014115001274333, 1.10533774173103271412725863539, 1.26503755453763160309475783615, 1.28264157129613450745864763285, 1.31873950076176200310164566330, 1.55295461496433297425722793148, 1.59360621940236050959034383426, 1.97332002234449329237824128486, 1.97401819198881136210101145532, 1.98986413176447260101797841218, 2.00153486797686611174814276530, 2.07532895581553336638670574120, 2.09081243277305316383276992632, 2.22550419246244089197561048668, 2.65561043657055664191972469328, 2.85970411739494850604020985321, 2.88990372677529900510762076423, 2.94517064562137940127330796954, 3.02010702653812905332638712944, 3.07513658022441424511644034182, 3.13782414757194344407617160456, 3.16508997064611439462640211727, 3.52179843108446757839385189578

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

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