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Properties

Label 20-2240e10-1.1-c1e10-0-1
Degree $20$
Conductor $3.180\times 10^{33}$
Sign $1$
Analytic cond. $3.35159\times 10^{12}$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 8·9-s − 8·11-s + 24·19-s + 5·25-s − 24·29-s + 24·31-s − 4·41-s + 16·45-s − 5·49-s − 16·55-s + 32·59-s + 20·61-s + 8·71-s − 64·79-s + 22·81-s − 4·89-s + 48·95-s − 64·99-s − 4·101-s − 8·109-s − 12·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s + 8/3·9-s − 2.41·11-s + 5.50·19-s + 25-s − 4.45·29-s + 4.31·31-s − 0.624·41-s + 2.38·45-s − 5/7·49-s − 2.15·55-s + 4.16·59-s + 2.56·61-s + 0.949·71-s − 7.20·79-s + 22/9·81-s − 0.423·89-s + 4.92·95-s − 6.43·99-s − 0.398·101-s − 0.766·109-s − 1.09·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{10} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{60} \cdot 5^{10} \cdot 7^{10}\)
Sign: $1$
Analytic conductor: \(3.35159\times 10^{12}\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{60} \cdot 5^{10} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.903287803\)
\(L(\frac12)\) \(\approx\) \(5.903287803\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T - T^{2} + 4 T^{3} + 4 p T^{4} - 116 T^{5} + 4 p^{2} T^{6} + 4 p^{2} T^{7} - p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
7 \( ( 1 + T^{2} )^{5} \)
good3 \( 1 - 8 T^{2} + 14 p T^{4} - 134 T^{6} + 115 p T^{8} - 860 T^{10} + 115 p^{3} T^{12} - 134 p^{4} T^{14} + 14 p^{7} T^{16} - 8 p^{8} T^{18} + p^{10} T^{20} \)
11 \( ( 1 + 4 T + 30 T^{2} + 24 T^{3} + 137 T^{4} - 568 T^{5} + 137 p T^{6} + 24 p^{2} T^{7} + 30 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
13 \( 1 - 36 T^{2} + 794 T^{4} - 8170 T^{6} + 43337 T^{8} + 138644 T^{10} + 43337 p^{2} T^{12} - 8170 p^{4} T^{14} + 794 p^{6} T^{16} - 36 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 - 104 T^{2} + 4966 T^{4} - 8654 p T^{6} + 3179161 T^{8} - 57319508 T^{10} + 3179161 p^{2} T^{12} - 8654 p^{5} T^{14} + 4966 p^{6} T^{16} - 104 p^{8} T^{18} + p^{10} T^{20} \)
19 \( ( 1 - 12 T + 125 T^{2} - 840 T^{3} + 5136 T^{4} - 23512 T^{5} + 5136 p T^{6} - 840 p^{2} T^{7} + 125 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
23 \( 1 - 126 T^{2} + 7501 T^{4} - 278920 T^{6} + 7632578 T^{8} - 180621172 T^{10} + 7632578 p^{2} T^{12} - 278920 p^{4} T^{14} + 7501 p^{6} T^{16} - 126 p^{8} T^{18} + p^{10} T^{20} \)
29 \( ( 1 + 12 T + 130 T^{2} + 930 T^{3} + 6981 T^{4} + 38500 T^{5} + 6981 p T^{6} + 930 p^{2} T^{7} + 130 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( ( 1 - 12 T + 99 T^{2} - 304 T^{3} - 78 T^{4} + 8312 T^{5} - 78 p T^{6} - 304 p^{2} T^{7} + 99 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
37 \( 1 - 234 T^{2} + 26997 T^{4} - 2045336 T^{6} + 113067106 T^{8} - 4763308988 T^{10} + 113067106 p^{2} T^{12} - 2045336 p^{4} T^{14} + 26997 p^{6} T^{16} - 234 p^{8} T^{18} + p^{10} T^{20} \)
41 \( ( 1 + 2 T + 105 T^{2} - 304 T^{3} + 3454 T^{4} - 31780 T^{5} + 3454 p T^{6} - 304 p^{2} T^{7} + 105 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( 1 - 2 p T^{2} + 5909 T^{4} - 183272 T^{6} + 2704034 T^{8} + 27726908 T^{10} + 2704034 p^{2} T^{12} - 183272 p^{4} T^{14} + 5909 p^{6} T^{16} - 2 p^{9} T^{18} + p^{10} T^{20} \)
47 \( 1 - 348 T^{2} + 58150 T^{4} - 6145418 T^{6} + 455711849 T^{8} - 24823499924 T^{10} + 455711849 p^{2} T^{12} - 6145418 p^{4} T^{14} + 58150 p^{6} T^{16} - 348 p^{8} T^{18} + p^{10} T^{20} \)
53 \( 1 - 322 T^{2} + 52133 T^{4} - 5598872 T^{6} + 440739922 T^{8} - 26538254604 T^{10} + 440739922 p^{2} T^{12} - 5598872 p^{4} T^{14} + 52133 p^{6} T^{16} - 322 p^{8} T^{18} + p^{10} T^{20} \)
59 \( ( 1 - 16 T + 189 T^{2} - 1728 T^{3} + 18280 T^{4} - 150624 T^{5} + 18280 p T^{6} - 1728 p^{2} T^{7} + 189 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
61 \( ( 1 - 10 T + 259 T^{2} - 1972 T^{3} + 28864 T^{4} - 167812 T^{5} + 28864 p T^{6} - 1972 p^{2} T^{7} + 259 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( 1 - 438 T^{2} + 88421 T^{4} - 11157864 T^{6} + 1022278818 T^{8} - 75045195140 T^{10} + 1022278818 p^{2} T^{12} - 11157864 p^{4} T^{14} + 88421 p^{6} T^{16} - 438 p^{8} T^{18} + p^{10} T^{20} \)
71 \( ( 1 - 4 T + 127 T^{2} + 208 T^{3} - 394 T^{4} + 70888 T^{5} - 394 p T^{6} + 208 p^{2} T^{7} + 127 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 - 506 T^{2} + 122781 T^{4} - 18992760 T^{6} + 2098764338 T^{8} - 174925478812 T^{10} + 2098764338 p^{2} T^{12} - 18992760 p^{4} T^{14} + 122781 p^{6} T^{16} - 506 p^{8} T^{18} + p^{10} T^{20} \)
79 \( ( 1 + 32 T + 758 T^{2} + 11996 T^{3} + 152941 T^{4} + 1499848 T^{5} + 152941 p T^{6} + 11996 p^{2} T^{7} + 758 p^{3} T^{8} + 32 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
83 \( 1 - 146 T^{2} + 11641 T^{4} - 1523040 T^{6} + 125145358 T^{8} - 6371959132 T^{10} + 125145358 p^{2} T^{12} - 1523040 p^{4} T^{14} + 11641 p^{6} T^{16} - 146 p^{8} T^{18} + p^{10} T^{20} \)
89 \( ( 1 + 2 T + 277 T^{2} + 312 T^{3} + 36290 T^{4} + 26956 T^{5} + 36290 p T^{6} + 312 p^{2} T^{7} + 277 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
97 \( 1 - 696 T^{2} + 237462 T^{4} - 51788462 T^{6} + 7950605161 T^{8} - 895242632756 T^{10} + 7950605161 p^{2} T^{12} - 51788462 p^{4} T^{14} + 237462 p^{6} T^{16} - 696 p^{8} T^{18} + p^{10} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.14620079587719546611015503666, −3.14003926862301740561915466118, −3.03181153939111944782451890274, −2.76092098946524273936329291759, −2.75428563485417695094831362917, −2.64604341662389529886152215425, −2.61646903385186984852304792949, −2.51946704953069280265439512749, −2.49691835337933546621075926051, −2.17961058780492638334941723488, −2.10300663019571400571790258334, −2.02749991973748905901777130889, −1.80952818000732024529823625946, −1.77395644868355699176337248226, −1.77266963744978042878225665795, −1.40733213691693121580229138562, −1.34655360401957525567917958476, −1.32176230789223831321836259246, −1.22140794721375206328617280436, −1.02609402615512245878091045794, −0.829180366314233630554997712580, −0.828240065267580172211029408471, −0.60856026000982959673843974147, −0.45542765497698412819413678186, −0.099096784617955763800732141512, 0.099096784617955763800732141512, 0.45542765497698412819413678186, 0.60856026000982959673843974147, 0.828240065267580172211029408471, 0.829180366314233630554997712580, 1.02609402615512245878091045794, 1.22140794721375206328617280436, 1.32176230789223831321836259246, 1.34655360401957525567917958476, 1.40733213691693121580229138562, 1.77266963744978042878225665795, 1.77395644868355699176337248226, 1.80952818000732024529823625946, 2.02749991973748905901777130889, 2.10300663019571400571790258334, 2.17961058780492638334941723488, 2.49691835337933546621075926051, 2.51946704953069280265439512749, 2.61646903385186984852304792949, 2.64604341662389529886152215425, 2.75428563485417695094831362917, 2.76092098946524273936329291759, 3.03181153939111944782451890274, 3.14003926862301740561915466118, 3.14620079587719546611015503666

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

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