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Properties

Label 16-310e8-1.1-c1e8-0-7
Degree $16$
Conductor $8.529\times 10^{19}$
Sign $1$
Analytic cond. $1409.64$
Root an. cond. $1.57332$
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  − 4·4-s − 2·5-s + 12·9-s + 12·11-s + 10·16-s − 4·19-s + 8·20-s + 8·29-s + 8·31-s − 48·36-s − 8·41-s − 48·44-s − 24·45-s + 28·49-s − 24·55-s − 20·64-s + 24·71-s + 16·76-s + 24·79-s − 20·80-s + 64·81-s + 24·89-s + 8·95-s + 144·99-s − 20·101-s + 12·109-s − 32·116-s + ⋯
L(s)  = 1  − 2·4-s − 0.894·5-s + 4·9-s + 3.61·11-s + 5/2·16-s − 0.917·19-s + 1.78·20-s + 1.48·29-s + 1.43·31-s − 8·36-s − 1.24·41-s − 7.23·44-s − 3.57·45-s + 4·49-s − 3.23·55-s − 5/2·64-s + 2.84·71-s + 1.83·76-s + 2.70·79-s − 2.23·80-s + 64/9·81-s + 2.54·89-s + 0.820·95-s + 14.4·99-s − 1.99·101-s + 1.14·109-s − 2.97·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(1409.64\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 5^{8} \cdot 31^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.196442913\)
\(L(\frac12)\) \(\approx\) \(4.196442913\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{4} \)
5 \( 1 + 2 T + 4 T^{2} + 6 T^{3} - 18 T^{4} + 6 p T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31 \( ( 1 - T )^{8} \)
good3 \( 1 - 4 p T^{2} + 80 T^{4} - 368 T^{6} + 1258 T^{8} - 368 p^{2} T^{10} + 80 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
7 \( 1 - 4 p T^{2} + 300 T^{4} - 1444 T^{6} + 5238 T^{8} - 1444 p^{2} T^{10} + 300 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 6 T + 50 T^{2} - 188 T^{3} + 856 T^{4} - 188 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 64 T^{2} + 2160 T^{4} - 47260 T^{6} + 726618 T^{8} - 47260 p^{2} T^{10} + 2160 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 112 T^{2} + 5796 T^{4} - 181120 T^{6} + 3741606 T^{8} - 181120 p^{2} T^{10} + 5796 p^{4} T^{12} - 112 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 2 T + 56 T^{2} + 50 T^{3} + 1366 T^{4} + 50 p T^{5} + 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 80 T^{2} + 3092 T^{4} - 93536 T^{6} + 2419654 T^{8} - 93536 p^{2} T^{10} + 3092 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 4 T + 82 T^{2} - 150 T^{3} + 2824 T^{4} - 150 p T^{5} + 82 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 204 T^{2} + 20656 T^{4} - 1324592 T^{6} + 58470586 T^{8} - 1324592 p^{2} T^{10} + 20656 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 4 T + 108 T^{2} + 608 T^{3} + 5458 T^{4} + 608 p T^{5} + 108 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 176 T^{2} + 16576 T^{4} - 1092588 T^{6} + 54065386 T^{8} - 1092588 p^{2} T^{10} + 16576 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 4 p T^{2} + 16108 T^{4} - 892164 T^{6} + 42114742 T^{8} - 892164 p^{2} T^{10} + 16108 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
53 \( 1 - 128 T^{2} + 6640 T^{4} - 451612 T^{6} + 620946 p T^{8} - 451612 p^{2} T^{10} + 6640 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 60 T^{2} - 608 T^{3} + 438 T^{4} - 608 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 106 T^{2} - 338 T^{3} + 7432 T^{4} - 338 p T^{5} + 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( 1 - 244 T^{2} + 25860 T^{4} - 1726060 T^{6} + 105932118 T^{8} - 1726060 p^{2} T^{10} + 25860 p^{4} T^{12} - 244 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 12 T + 188 T^{2} - 1916 T^{3} + 20118 T^{4} - 1916 p T^{5} + 188 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 192 T^{2} + 28340 T^{4} - 2905328 T^{6} + 242721478 T^{8} - 2905328 p^{2} T^{10} + 28340 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 12 T + 348 T^{2} - 2812 T^{3} + 42422 T^{4} - 2812 p T^{5} + 348 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 356 T^{2} + 68096 T^{4} - 8819048 T^{6} + 842939626 T^{8} - 8819048 p^{2} T^{10} + 68096 p^{4} T^{12} - 356 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 12 T + 116 T^{2} + 188 T^{3} - 954 T^{4} + 188 p T^{5} + 116 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 12 T^{2} + 29020 T^{4} + 247508 T^{6} + 383961238 T^{8} + 247508 p^{2} T^{10} + 29020 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} \)
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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

−5.06714257986021403749122812034, −4.94588483357517826607584749653, −4.80892645452724957872668366752, −4.71883319792441897598133474416, −4.56023725220874372782715110565, −4.48922672463026160604374198871, −4.17779458516582549589139341537, −4.14063734308090964011070777509, −4.03163610530703696568759421429, −3.96386371826481454815970419483, −3.84655997027808174360021860771, −3.73905994944898685905500697611, −3.59040356676620555176075355836, −3.35622130248904037507948230898, −3.16951795692188074473803437244, −2.84808376295958193657944876939, −2.65236140839397967119995475184, −2.17376073749496433723391921040, −2.11765606463432876756980700565, −1.88514707584235547361292973645, −1.65444680949152267990034502378, −1.21209082254864305147923445681, −1.10286945066020886871114462058, −0.873211372093481929193817551699, −0.797071585431267279857302228674, 0.797071585431267279857302228674, 0.873211372093481929193817551699, 1.10286945066020886871114462058, 1.21209082254864305147923445681, 1.65444680949152267990034502378, 1.88514707584235547361292973645, 2.11765606463432876756980700565, 2.17376073749496433723391921040, 2.65236140839397967119995475184, 2.84808376295958193657944876939, 3.16951795692188074473803437244, 3.35622130248904037507948230898, 3.59040356676620555176075355836, 3.73905994944898685905500697611, 3.84655997027808174360021860771, 3.96386371826481454815970419483, 4.03163610530703696568759421429, 4.14063734308090964011070777509, 4.17779458516582549589139341537, 4.48922672463026160604374198871, 4.56023725220874372782715110565, 4.71883319792441897598133474416, 4.80892645452724957872668366752, 4.94588483357517826607584749653, 5.06714257986021403749122812034

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

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