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Properties

Label 20-296e10-1.1-c1e10-0-0
Degree $20$
Conductor $5.163\times 10^{24}$
Sign $1$
Analytic cond. $5441.11$
Root an. cond. $1.53739$
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  + 2·3-s − 4·7-s − 7·9-s + 2·11-s − 8·21-s + 27·25-s − 16·27-s + 4·33-s − 6·37-s − 26·41-s + 8·47-s − 20·49-s − 20·53-s + 28·63-s − 2·67-s − 4·71-s − 14·73-s + 54·75-s − 8·77-s + 20·81-s − 36·83-s − 14·99-s − 36·101-s − 10·107-s − 12·111-s − 51·121-s − 52·123-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.51·7-s − 7/3·9-s + 0.603·11-s − 1.74·21-s + 27/5·25-s − 3.07·27-s + 0.696·33-s − 0.986·37-s − 4.06·41-s + 1.16·47-s − 2.85·49-s − 2.74·53-s + 3.52·63-s − 0.244·67-s − 0.474·71-s − 1.63·73-s + 6.23·75-s − 0.911·77-s + 20/9·81-s − 3.95·83-s − 1.40·99-s − 3.58·101-s − 0.966·107-s − 1.13·111-s − 4.63·121-s − 4.68·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 37^{10}\)
Sign: $1$
Analytic conductor: \(5441.11\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 37^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.453969922\)
\(L(\frac12)\) \(\approx\) \(1.453969922\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + 6 T + p T^{2} + 256 T^{3} + 54 p T^{4} + 580 p T^{5} + 54 p^{2} T^{6} + 256 p^{2} T^{7} + p^{4} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
good3 \( ( 1 - T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - 8 T^{5} + 5 p^{2} T^{6} - 5 p^{2} T^{7} + 5 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
5 \( 1 - 27 T^{2} + 382 T^{4} - 3674 T^{6} + 26561 T^{8} - 149654 T^{10} + 26561 p^{2} T^{12} - 3674 p^{4} T^{14} + 382 p^{6} T^{16} - 27 p^{8} T^{18} + p^{10} T^{20} \)
7 \( ( 1 + 2 T + 16 T^{2} + 36 T^{3} + 25 p T^{4} + 276 T^{5} + 25 p^{2} T^{6} + 36 p^{2} T^{7} + 16 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
11 \( ( 1 - T + 27 T^{2} - 81 T^{3} + 359 T^{4} - 1424 T^{5} + 359 p T^{6} - 81 p^{2} T^{7} + 27 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \)
13 \( 1 - 75 T^{2} + 2638 T^{4} - 60202 T^{6} + 1041777 T^{8} - 14787478 T^{10} + 1041777 p^{2} T^{12} - 60202 p^{4} T^{14} + 2638 p^{6} T^{16} - 75 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 - 86 T^{2} + 3949 T^{4} - 123400 T^{6} + 172690 p T^{8} - 55504452 T^{10} + 172690 p^{3} T^{12} - 123400 p^{4} T^{14} + 3949 p^{6} T^{16} - 86 p^{8} T^{18} + p^{10} T^{20} \)
19 \( 1 - 102 T^{2} + 5189 T^{4} - 180072 T^{6} + 4786338 T^{8} - 101417252 T^{10} + 4786338 p^{2} T^{12} - 180072 p^{4} T^{14} + 5189 p^{6} T^{16} - 102 p^{8} T^{18} + p^{10} T^{20} \)
23 \( 1 - 59 T^{2} + 2362 T^{4} - 71938 T^{6} + 1780517 T^{8} - 41719878 T^{10} + 1780517 p^{2} T^{12} - 71938 p^{4} T^{14} + 2362 p^{6} T^{16} - 59 p^{8} T^{18} + p^{10} T^{20} \)
29 \( 1 - 83 T^{2} + 5110 T^{4} - 233410 T^{6} + 8977177 T^{8} - 278520182 T^{10} + 8977177 p^{2} T^{12} - 233410 p^{4} T^{14} + 5110 p^{6} T^{16} - 83 p^{8} T^{18} + p^{10} T^{20} \)
31 \( 1 - 87 T^{2} + 5802 T^{4} - 274882 T^{6} + 10790773 T^{8} - 360535438 T^{10} + 10790773 p^{2} T^{12} - 274882 p^{4} T^{14} + 5802 p^{6} T^{16} - 87 p^{8} T^{18} + p^{10} T^{20} \)
41 \( ( 1 + 13 T + 209 T^{2} + 1939 T^{3} + 17003 T^{4} + 115174 T^{5} + 17003 p T^{6} + 1939 p^{2} T^{7} + 209 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( 1 - 234 T^{2} + 27269 T^{4} - 2176312 T^{6} + 132775346 T^{8} - 6400847740 T^{10} + 132775346 p^{2} T^{12} - 2176312 p^{4} T^{14} + 27269 p^{6} T^{16} - 234 p^{8} T^{18} + p^{10} T^{20} \)
47 \( ( 1 - 4 T + 84 T^{2} - 698 T^{3} + 5811 T^{4} - 36356 T^{5} + 5811 p T^{6} - 698 p^{2} T^{7} + 84 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
53 \( ( 1 + 10 T + 242 T^{2} + 1556 T^{3} + 22361 T^{4} + 106492 T^{5} + 22361 p T^{6} + 1556 p^{2} T^{7} + 242 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( 1 - 206 T^{2} + 24805 T^{4} - 2226472 T^{6} + 161427122 T^{8} - 10084331604 T^{10} + 161427122 p^{2} T^{12} - 2226472 p^{4} T^{14} + 24805 p^{6} T^{16} - 206 p^{8} T^{18} + p^{10} T^{20} \)
61 \( 1 - 283 T^{2} + 37950 T^{4} - 3603250 T^{6} + 293158289 T^{8} - 20024621830 T^{10} + 293158289 p^{2} T^{12} - 3603250 p^{4} T^{14} + 37950 p^{6} T^{16} - 283 p^{8} T^{18} + p^{10} T^{20} \)
67 \( ( 1 + T + 172 T^{2} + 288 T^{3} + 13903 T^{4} + 27246 T^{5} + 13903 p T^{6} + 288 p^{2} T^{7} + 172 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \)
71 \( ( 1 + 2 T + 160 T^{2} - 608 T^{3} + 10055 T^{4} - 95204 T^{5} + 10055 p T^{6} - 608 p^{2} T^{7} + 160 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( ( 1 + 7 T + 269 T^{2} + 1525 T^{3} + 34743 T^{4} + 155902 T^{5} + 34743 p T^{6} + 1525 p^{2} T^{7} + 269 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
79 \( 1 - 459 T^{2} + 87882 T^{4} - 8843178 T^{6} + 505513637 T^{8} - 26547028982 T^{10} + 505513637 p^{2} T^{12} - 8843178 p^{4} T^{14} + 87882 p^{6} T^{16} - 459 p^{8} T^{18} + p^{10} T^{20} \)
83 \( ( 1 + 18 T + 384 T^{2} + 4984 T^{3} + 57619 T^{4} + 575564 T^{5} + 57619 p T^{6} + 4984 p^{2} T^{7} + 384 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( 1 - 410 T^{2} + 104637 T^{4} - 17962488 T^{6} + 2355052850 T^{8} - 235651627612 T^{10} + 2355052850 p^{2} T^{12} - 17962488 p^{4} T^{14} + 104637 p^{6} T^{16} - 410 p^{8} T^{18} + p^{10} T^{20} \)
97 \( 1 - 254 T^{2} + 43549 T^{4} - 5636232 T^{6} + 662391202 T^{8} - 68393075956 T^{10} + 662391202 p^{2} T^{12} - 5636232 p^{4} T^{14} + 43549 p^{6} T^{16} - 254 p^{8} T^{18} + p^{10} T^{20} \)
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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

−4.59814935992063566343153686191, −4.52126739851937292145703771364, −4.17015089272688981969427592947, −4.12240627800310829549929391085, −4.05174555835615803957436033442, −4.04608449418846179171438392918, −3.58364128670277834573251986020, −3.40927556951468858727905975895, −3.28126948184355467341884439066, −3.13271647476729842552418210739, −3.12186639027770530718832129651, −3.08179430255719326701690429168, −3.07724444741790325675617359648, −2.91094608570265359725080843086, −2.79772465964524725255718398330, −2.69768073394493057799427604278, −2.52502684961688090903415891854, −2.09764002429421684948792833945, −1.76111964409577702655786827316, −1.71235094011254318496823660981, −1.60422875976813798341277613695, −1.58318803682030761852413160587, −1.08121727882142999428888668292, −0.52666618835100141805554125024, −0.34004613081877644673656560276, 0.34004613081877644673656560276, 0.52666618835100141805554125024, 1.08121727882142999428888668292, 1.58318803682030761852413160587, 1.60422875976813798341277613695, 1.71235094011254318496823660981, 1.76111964409577702655786827316, 2.09764002429421684948792833945, 2.52502684961688090903415891854, 2.69768073394493057799427604278, 2.79772465964524725255718398330, 2.91094608570265359725080843086, 3.07724444741790325675617359648, 3.08179430255719326701690429168, 3.12186639027770530718832129651, 3.13271647476729842552418210739, 3.28126948184355467341884439066, 3.40927556951468858727905975895, 3.58364128670277834573251986020, 4.04608449418846179171438392918, 4.05174555835615803957436033442, 4.12240627800310829549929391085, 4.17015089272688981969427592947, 4.52126739851937292145703771364, 4.59814935992063566343153686191

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

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