Properties

Label 14-5577e7-1.1-c1e7-0-0
Degree $14$
Conductor $1.678\times 10^{26}$
Sign $1$
Analytic cond. $3.47331\times 10^{11}$
Root an. cond. $6.67327$
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  + 3·2-s + 7·3-s + 2·4-s + 6·5-s + 21·6-s + 6·7-s + 28·9-s + 18·10-s + 7·11-s + 14·12-s + 18·14-s + 42·15-s + 4·16-s − 2·17-s + 84·18-s + 8·19-s + 12·20-s + 42·21-s + 21·22-s + 4·23-s + 7·25-s + 84·27-s + 12·28-s − 12·29-s + 126·30-s − 10·31-s + 6·32-s + ⋯
L(s)  = 1  + 2.12·2-s + 4.04·3-s + 4-s + 2.68·5-s + 8.57·6-s + 2.26·7-s + 28/3·9-s + 5.69·10-s + 2.11·11-s + 4.04·12-s + 4.81·14-s + 10.8·15-s + 16-s − 0.485·17-s + 19.7·18-s + 1.83·19-s + 2.68·20-s + 9.16·21-s + 4.47·22-s + 0.834·23-s + 7/5·25-s + 16.1·27-s + 2.26·28-s − 2.22·29-s + 23.0·30-s − 1.79·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 11^{7} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 11^{7} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(3^{7} \cdot 11^{7} \cdot 13^{14}\)
Sign: $1$
Analytic conductor: \(3.47331\times 10^{11}\)
Root analytic conductor: \(6.67327\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5577} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 3^{7} \cdot 11^{7} \cdot 13^{14} ,\ ( \ : [1/2]^{7} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2747.051821\)
\(L(\frac12)\) \(\approx\) \(2747.051821\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{7} \)
11 \( ( 1 - T )^{7} \)
13 \( 1 \)
good2 \( 1 - 3 T + 7 T^{2} - 15 T^{3} + 27 T^{4} - 45 T^{5} + 71 T^{6} - 101 T^{7} + 71 p T^{8} - 45 p^{2} T^{9} + 27 p^{3} T^{10} - 15 p^{4} T^{11} + 7 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
5 \( 1 - 6 T + 29 T^{2} - 106 T^{3} + 343 T^{4} - 968 T^{5} + 2539 T^{6} - 5888 T^{7} + 2539 p T^{8} - 968 p^{2} T^{9} + 343 p^{3} T^{10} - 106 p^{4} T^{11} + 29 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
7 \( 1 - 6 T + 31 T^{2} - 116 T^{3} + 383 T^{4} - 1126 T^{5} + 3009 T^{6} - 8016 T^{7} + 3009 p T^{8} - 1126 p^{2} T^{9} + 383 p^{3} T^{10} - 116 p^{4} T^{11} + 31 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 2 T + 69 T^{2} + 172 T^{3} + 2517 T^{4} + 6628 T^{5} + 60381 T^{6} + 144332 T^{7} + 60381 p T^{8} + 6628 p^{2} T^{9} + 2517 p^{3} T^{10} + 172 p^{4} T^{11} + 69 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 - 8 T + 111 T^{2} - 744 T^{3} + 5735 T^{4} - 31188 T^{5} + 174057 T^{6} - 757800 T^{7} + 174057 p T^{8} - 31188 p^{2} T^{9} + 5735 p^{3} T^{10} - 744 p^{4} T^{11} + 111 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 4 T + 85 T^{2} - 504 T^{3} + 3817 T^{4} - 25280 T^{5} + 123013 T^{6} - 728008 T^{7} + 123013 p T^{8} - 25280 p^{2} T^{9} + 3817 p^{3} T^{10} - 504 p^{4} T^{11} + 85 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 12 T + 205 T^{2} + 1730 T^{3} + 17161 T^{4} + 111022 T^{5} + 804745 T^{6} + 4114856 T^{7} + 804745 p T^{8} + 111022 p^{2} T^{9} + 17161 p^{3} T^{10} + 1730 p^{4} T^{11} + 205 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 + 10 T + 5 p T^{2} + 1266 T^{3} + 11031 T^{4} + 74908 T^{5} + 483933 T^{6} + 2807256 T^{7} + 483933 p T^{8} + 74908 p^{2} T^{9} + 11031 p^{3} T^{10} + 1266 p^{4} T^{11} + 5 p^{6} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 6 T + 123 T^{2} - 540 T^{3} + 8341 T^{4} - 33890 T^{5} + 429655 T^{6} - 1511352 T^{7} + 429655 p T^{8} - 33890 p^{2} T^{9} + 8341 p^{3} T^{10} - 540 p^{4} T^{11} + 123 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 - 2 T + 157 T^{2} - 312 T^{3} + 13071 T^{4} - 24834 T^{5} + 731507 T^{6} - 1223368 T^{7} + 731507 p T^{8} - 24834 p^{2} T^{9} + 13071 p^{3} T^{10} - 312 p^{4} T^{11} + 157 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 + 16 T + 211 T^{2} + 2288 T^{3} + 22441 T^{4} + 191142 T^{5} + 1465639 T^{6} + 9828708 T^{7} + 1465639 p T^{8} + 191142 p^{2} T^{9} + 22441 p^{3} T^{10} + 2288 p^{4} T^{11} + 211 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 - 18 T + 285 T^{2} - 2340 T^{3} + 15929 T^{4} - 31606 T^{5} - 173731 T^{6} + 3507160 T^{7} - 173731 p T^{8} - 31606 p^{2} T^{9} + 15929 p^{3} T^{10} - 2340 p^{4} T^{11} + 285 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 - 10 T + 283 T^{2} - 1932 T^{3} + 32981 T^{4} - 164822 T^{5} + 2330967 T^{6} - 9583240 T^{7} + 2330967 p T^{8} - 164822 p^{2} T^{9} + 32981 p^{3} T^{10} - 1932 p^{4} T^{11} + 283 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 - 2 T + 253 T^{2} + 12 T^{3} + 30397 T^{4} + 41266 T^{5} + 2394233 T^{6} + 4109736 T^{7} + 2394233 p T^{8} + 41266 p^{2} T^{9} + 30397 p^{3} T^{10} + 12 p^{4} T^{11} + 253 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 + 10 T + 439 T^{2} + 3544 T^{3} + 81553 T^{4} + 529902 T^{5} + 8345663 T^{6} + 42813248 T^{7} + 8345663 p T^{8} + 529902 p^{2} T^{9} + 81553 p^{3} T^{10} + 3544 p^{4} T^{11} + 439 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - 8 T + 247 T^{2} - 1606 T^{3} + 34415 T^{4} - 2822 p T^{5} + 3206121 T^{6} - 15229160 T^{7} + 3206121 p T^{8} - 2822 p^{3} T^{9} + 34415 p^{3} T^{10} - 1606 p^{4} T^{11} + 247 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 36 T + 849 T^{2} - 14216 T^{3} + 197573 T^{4} - 2288676 T^{5} + 23379653 T^{6} - 208029408 T^{7} + 23379653 p T^{8} - 2288676 p^{2} T^{9} + 197573 p^{3} T^{10} - 14216 p^{4} T^{11} + 849 p^{5} T^{12} - 36 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 - 20 T + 397 T^{2} - 4216 T^{3} + 42675 T^{4} - 266264 T^{5} + 1819487 T^{6} - 10056872 T^{7} + 1819487 p T^{8} - 266264 p^{2} T^{9} + 42675 p^{3} T^{10} - 4216 p^{4} T^{11} + 397 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 - 6 T + 399 T^{2} - 2284 T^{3} + 76961 T^{4} - 389888 T^{5} + 9195259 T^{6} - 38954460 T^{7} + 9195259 p T^{8} - 389888 p^{2} T^{9} + 76961 p^{3} T^{10} - 2284 p^{4} T^{11} + 399 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 - 30 T + 813 T^{2} - 14236 T^{3} + 224949 T^{4} - 2797346 T^{5} + 31904401 T^{6} - 302487208 T^{7} + 31904401 p T^{8} - 2797346 p^{2} T^{9} + 224949 p^{3} T^{10} - 14236 p^{4} T^{11} + 813 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 - 34 T + 953 T^{2} - 18368 T^{3} + 304551 T^{4} - 4090348 T^{5} + 48525519 T^{6} - 485121860 T^{7} + 48525519 p T^{8} - 4090348 p^{2} T^{9} + 304551 p^{3} T^{10} - 18368 p^{4} T^{11} + 953 p^{5} T^{12} - 34 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 16 T + 467 T^{2} - 4096 T^{3} + 65633 T^{4} - 203976 T^{5} + 3647819 T^{6} + 8084944 T^{7} + 3647819 p T^{8} - 203976 p^{2} T^{9} + 65633 p^{3} T^{10} - 4096 p^{4} T^{11} + 467 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.65118402700268653646330611729, −3.64933768369989421152563703771, −3.54711883129777226200809327271, −3.48879313026094751394353694059, −3.16200787216044213230439990459, −3.11162645489294368959601000460, −3.06008488995993502544163031774, −3.02585224490340776570661532012, −2.67266729612665097313494257753, −2.44519818687023432966200326740, −2.24981988016235072659071400835, −2.24030896379559241169755955226, −2.20191019366229916208339818894, −2.10870687093367246785529891569, −1.95411892567305961799573931540, −1.85454842283752599940188115452, −1.79222977389561937264660185332, −1.60769981803563766395743158731, −1.58417792399920402609523445120, −1.36815985258767532950655311011, −1.23943338362542005490732876099, −0.819726463289637517392794895211, −0.75622406698047440514594154983, −0.74649164071958294395262759525, −0.56615957654281810379048850305, 0.56615957654281810379048850305, 0.74649164071958294395262759525, 0.75622406698047440514594154983, 0.819726463289637517392794895211, 1.23943338362542005490732876099, 1.36815985258767532950655311011, 1.58417792399920402609523445120, 1.60769981803563766395743158731, 1.79222977389561937264660185332, 1.85454842283752599940188115452, 1.95411892567305961799573931540, 2.10870687093367246785529891569, 2.20191019366229916208339818894, 2.24030896379559241169755955226, 2.24981988016235072659071400835, 2.44519818687023432966200326740, 2.67266729612665097313494257753, 3.02585224490340776570661532012, 3.06008488995993502544163031774, 3.11162645489294368959601000460, 3.16200787216044213230439990459, 3.48879313026094751394353694059, 3.54711883129777226200809327271, 3.64933768369989421152563703771, 3.65118402700268653646330611729

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

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