Properties

Label 16-1950e8-1.1-c1e8-0-0
Degree $16$
Conductor $2.091\times 10^{26}$
Sign $1$
Analytic cond. $3.45536\times 10^{9}$
Root an. cond. $3.94598$
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  + 2·4-s + 2·9-s − 6·11-s + 16-s − 18·19-s − 12·29-s + 32·31-s + 4·36-s − 4·41-s − 12·44-s − 15·49-s − 8·59-s − 18·61-s − 2·64-s − 42·71-s − 36·76-s + 44·79-s + 81-s + 8·89-s − 12·99-s − 8·101-s − 60·109-s − 24·116-s + 49·121-s + 64·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 4-s + 2/3·9-s − 1.80·11-s + 1/4·16-s − 4.12·19-s − 2.22·29-s + 5.74·31-s + 2/3·36-s − 0.624·41-s − 1.80·44-s − 2.14·49-s − 1.04·59-s − 2.30·61-s − 1/4·64-s − 4.98·71-s − 4.12·76-s + 4.95·79-s + 1/9·81-s + 0.847·89-s − 1.20·99-s − 0.796·101-s − 5.74·109-s − 2.22·116-s + 4.45·121-s + 5.74·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(3.45536\times 10^{9}\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.0002902790396\)
\(L(\frac12)\) \(\approx\) \(0.0002902790396\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{2} \)
3 \( ( 1 - T^{2} + T^{4} )^{2} \)
5 \( 1 \)
13 \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
good7 \( 1 + 15 T^{2} + 109 T^{4} + 270 T^{6} + 30 T^{8} + 270 p^{2} T^{10} + 109 p^{4} T^{12} + 15 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 3 T - p T^{2} - 6 T^{3} + 180 T^{4} - 6 p T^{5} - p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 + 32 T^{2} + 258 T^{4} + 6016 T^{6} + 207299 T^{8} + 6016 p^{2} T^{10} + 258 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 9 T + 27 T^{2} + 144 T^{3} + 1016 T^{4} + 144 p T^{5} + 27 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 11 T^{2} - 623 T^{4} - 3454 T^{6} + 209686 T^{8} - 3454 p^{2} T^{10} - 623 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 6 T - 14 T^{2} - 48 T^{3} + 615 T^{4} - 48 p T^{5} - 14 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 4 T + p T^{2} )^{8} \)
37 \( 1 + 135 T^{2} + 10969 T^{4} + 609930 T^{6} + 25825350 T^{8} + 609930 p^{2} T^{10} + 10969 p^{4} T^{12} + 135 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 2 T - 62 T^{2} - 32 T^{3} + 2511 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 + 151 T^{2} + 13509 T^{4} + 844694 T^{6} + 40763414 T^{8} + 844694 p^{2} T^{10} + 13509 p^{4} T^{12} + 151 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 44 T^{2} + 1750 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 4 T - 38 T^{2} - 256 T^{3} - 1509 T^{4} - 256 p T^{5} - 38 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 9 T - 23 T^{2} - 162 T^{3} + 2154 T^{4} - 162 p T^{5} - 23 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 51 T^{2} - 6071 T^{4} - 15606 T^{6} + 43376574 T^{8} - 15606 p^{2} T^{10} - 6071 p^{4} T^{12} + 51 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 21 T + 193 T^{2} + 2226 T^{3} + 25152 T^{4} + 2226 p T^{5} + 193 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 11 T + 82 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 283 T^{2} + 33456 T^{4} - 283 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 4 T - 98 T^{2} + 256 T^{3} + 3651 T^{4} + 256 p T^{5} - 98 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 63 T^{2} + 8065 T^{4} - 1443582 T^{6} - 109096386 T^{8} - 1443582 p^{2} T^{10} + 8065 p^{4} T^{12} + 63 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.87651038779460206804680874286, −3.63587097568809932978416494472, −3.62631570011762295806000923973, −3.53108375555297639513992927369, −3.28376177619596730221954981181, −3.23056223218469422349534041556, −2.94947859657399609313782352910, −2.90991550119530379275506482430, −2.86323950748070467264539124750, −2.81680502089314581568427393003, −2.47954388237358839281174878466, −2.45506879343894802214049667486, −2.29663838074397059155303100401, −2.24173247857248342261338654467, −2.02125978727253124704450263963, −1.90506020000540712714558383320, −1.86927099160588701305880007077, −1.66299714200670888082102438874, −1.35885925759338636711128654148, −1.27837314520625049062211265122, −1.07978222182576244529399088228, −1.03912591936287826653756121260, −0.53144022772354775096773434576, −0.26237931149579169125726880491, −0.00252568031842087777935157000, 0.00252568031842087777935157000, 0.26237931149579169125726880491, 0.53144022772354775096773434576, 1.03912591936287826653756121260, 1.07978222182576244529399088228, 1.27837314520625049062211265122, 1.35885925759338636711128654148, 1.66299714200670888082102438874, 1.86927099160588701305880007077, 1.90506020000540712714558383320, 2.02125978727253124704450263963, 2.24173247857248342261338654467, 2.29663838074397059155303100401, 2.45506879343894802214049667486, 2.47954388237358839281174878466, 2.81680502089314581568427393003, 2.86323950748070467264539124750, 2.90991550119530379275506482430, 2.94947859657399609313782352910, 3.23056223218469422349534041556, 3.28376177619596730221954981181, 3.53108375555297639513992927369, 3.62631570011762295806000923973, 3.63587097568809932978416494472, 3.87651038779460206804680874286

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

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