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Properties

Label 40-3168e20-1.1-c1e20-0-0
Degree $40$
Conductor $1.037\times 10^{70}$
Sign $1$
Analytic cond. $1.15145\times 10^{28}$
Root an. cond. $5.02957$
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  + 44·25-s − 40·31-s − 52·49-s − 56·73-s + 16·79-s + 8·97-s + 40·103-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 44/5·25-s − 7.18·31-s − 7.42·49-s − 6.55·73-s + 1.80·79-s + 0.812·97-s + 3.94·103-s − 0.909·121-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 + 8.30·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 3^{40} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 3^{40} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{100} \cdot 3^{40} \cdot 11^{20}\)
Sign: $1$
Analytic conductor: \(1.15145\times 10^{28}\)
Root analytic conductor: \(5.02957\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{100} \cdot 3^{40} \cdot 11^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.224701448\)
\(L(\frac12)\) \(\approx\) \(1.224701448\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( ( 1 + T^{2} )^{10} \)
good5 \( ( 1 - 22 T^{2} + 249 T^{4} - 384 p T^{6} + 11582 T^{8} - 60564 T^{10} + 11582 p^{2} T^{12} - 384 p^{5} T^{14} + 249 p^{6} T^{16} - 22 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
7 \( ( 1 + 13 T^{2} + 16 T^{3} + 68 T^{4} + 208 T^{5} + 68 p T^{6} + 16 p^{2} T^{7} + 13 p^{3} T^{8} + p^{5} T^{10} )^{4} \)
13 \( ( 1 - 54 T^{2} + 1641 T^{4} - 38176 T^{6} + 686494 T^{8} - 9812116 T^{10} + 686494 p^{2} T^{12} - 38176 p^{4} T^{14} + 1641 p^{6} T^{16} - 54 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
17 \( ( 1 + 54 T^{2} + 1933 T^{4} + 47816 T^{6} + 1027730 T^{8} + 18283268 T^{10} + 1027730 p^{2} T^{12} + 47816 p^{4} T^{14} + 1933 p^{6} T^{16} + 54 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
19 \( ( 1 - 82 T^{2} + 3685 T^{4} - 118776 T^{6} + 3036034 T^{8} - 63392172 T^{10} + 3036034 p^{2} T^{12} - 118776 p^{4} T^{14} + 3685 p^{6} T^{16} - 82 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
23 \( ( 1 + 90 T^{2} + 4593 T^{4} + 320 p^{2} T^{6} + 5160798 T^{8} + 130112588 T^{10} + 5160798 p^{2} T^{12} + 320 p^{6} T^{14} + 4593 p^{6} T^{16} + 90 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
29 \( ( 1 - 158 T^{2} + 13269 T^{4} - 758248 T^{6} + 32272242 T^{8} - 1059695028 T^{10} + 32272242 p^{2} T^{12} - 758248 p^{4} T^{14} + 13269 p^{6} T^{16} - 158 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
31 \( ( 1 + 10 T + 159 T^{2} + 1120 T^{3} + 9950 T^{4} + 50284 T^{5} + 9950 p T^{6} + 1120 p^{2} T^{7} + 159 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
37 \( ( 1 - 146 T^{2} + 10213 T^{4} - 531096 T^{6} + 25861234 T^{8} - 1083826540 T^{10} + 25861234 p^{2} T^{12} - 531096 p^{4} T^{14} + 10213 p^{6} T^{16} - 146 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
41 \( ( 1 + 222 T^{2} + 24877 T^{4} + 1853576 T^{6} + 103922786 T^{8} + 4690087604 T^{10} + 103922786 p^{2} T^{12} + 1853576 p^{4} T^{14} + 24877 p^{6} T^{16} + 222 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
43 \( ( 1 - 242 T^{2} + 29429 T^{4} - 2370296 T^{6} + 142481090 T^{8} - 6800931052 T^{10} + 142481090 p^{2} T^{12} - 2370296 p^{4} T^{14} + 29429 p^{6} T^{16} - 242 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
47 \( ( 1 + 122 T^{2} + 10305 T^{4} + 714304 T^{6} + 42467166 T^{8} + 2242451340 T^{10} + 42467166 p^{2} T^{12} + 714304 p^{4} T^{14} + 10305 p^{6} T^{16} + 122 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
53 \( ( 1 - 310 T^{2} + 49945 T^{4} - 5370112 T^{6} + 424321918 T^{8} - 25591088212 T^{10} + 424321918 p^{2} T^{12} - 5370112 p^{4} T^{14} + 49945 p^{6} T^{16} - 310 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
59 \( ( 1 - 382 T^{2} + 73317 T^{4} - 9214632 T^{6} + 833455154 T^{8} - 56478974964 T^{10} + 833455154 p^{2} T^{12} - 9214632 p^{4} T^{14} + 73317 p^{6} T^{16} - 382 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
61 \( ( 1 - 182 T^{2} + 19081 T^{4} - 1638944 T^{6} + 120234462 T^{8} - 7639419028 T^{10} + 120234462 p^{2} T^{12} - 1638944 p^{4} T^{14} + 19081 p^{6} T^{16} - 182 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
67 \( ( 1 - 262 T^{2} + 44773 T^{4} - 5313576 T^{6} + 496568098 T^{8} - 36806562660 T^{10} + 496568098 p^{2} T^{12} - 5313576 p^{4} T^{14} + 44773 p^{6} T^{16} - 262 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
71 \( ( 1 + 554 T^{2} + 146641 T^{4} + 24336960 T^{6} + 2796888478 T^{8} + 232213456300 T^{10} + 2796888478 p^{2} T^{12} + 24336960 p^{4} T^{14} + 146641 p^{6} T^{16} + 554 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
73 \( ( 1 + 14 T + 325 T^{2} + 3080 T^{3} + 42370 T^{4} + 300692 T^{5} + 42370 p T^{6} + 3080 p^{2} T^{7} + 325 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
79 \( ( 1 - 4 T + 309 T^{2} - 952 T^{3} + 43028 T^{4} - 100792 T^{5} + 43028 p T^{6} - 952 p^{2} T^{7} + 309 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
83 \( ( 1 - 358 T^{2} + 54085 T^{4} - 5744552 T^{6} + 643317794 T^{8} - 63390374052 T^{10} + 643317794 p^{2} T^{12} - 5744552 p^{4} T^{14} + 54085 p^{6} T^{16} - 358 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
89 \( ( 1 + 474 T^{2} + 112989 T^{4} + 18425016 T^{6} + 2294354834 T^{8} + 227494676060 T^{10} + 2294354834 p^{2} T^{12} + 18425016 p^{4} T^{14} + 112989 p^{6} T^{16} + 474 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
97 \( ( 1 - 2 T + 289 T^{2} + 192 T^{3} + 37438 T^{4} + 70020 T^{5} + 37438 p T^{6} + 192 p^{2} T^{7} + 289 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.73297069120401258636493103491, −1.66506265897131498263980236520, −1.66266295422843400004231852248, −1.61760758709161446932348950468, −1.51296397377198029706681033901, −1.50635019103201743594493037431, −1.41864145836190885568084068143, −1.37957959659335743087378773065, −1.33351497644065391150139023824, −1.20253250090748689558387990341, −1.18625425482447420152055770469, −1.17179453190661374330963415452, −1.11826907766279207720415416628, −0.877956350121003108404078294041, −0.856994129476758375000004471792, −0.834963780092712150856410184190, −0.78424492399056099691120569795, −0.62973512247178280739244459069, −0.60549946502168140593695129887, −0.54733277304014268864160669339, −0.42273307073727792251212856234, −0.27652975110918122732431732305, −0.20833674645520587901168161504, −0.086354631901584931141363955224, −0.079350604941791020611071695099, 0.079350604941791020611071695099, 0.086354631901584931141363955224, 0.20833674645520587901168161504, 0.27652975110918122732431732305, 0.42273307073727792251212856234, 0.54733277304014268864160669339, 0.60549946502168140593695129887, 0.62973512247178280739244459069, 0.78424492399056099691120569795, 0.834963780092712150856410184190, 0.856994129476758375000004471792, 0.877956350121003108404078294041, 1.11826907766279207720415416628, 1.17179453190661374330963415452, 1.18625425482447420152055770469, 1.20253250090748689558387990341, 1.33351497644065391150139023824, 1.37957959659335743087378773065, 1.41864145836190885568084068143, 1.50635019103201743594493037431, 1.51296397377198029706681033901, 1.61760758709161446932348950468, 1.66266295422843400004231852248, 1.66506265897131498263980236520, 1.73297069120401258636493103491

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

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