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Properties

Label 14-4030e7-1.1-c1e7-0-0
Degree $14$
Conductor $1.726\times 10^{25}$
Sign $-1$
Analytic cond. $3.57334\times 10^{10}$
Root an. cond. $5.67271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $7$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·2-s + 3·3-s + 28·4-s − 7·5-s − 21·6-s + 2·7-s − 84·8-s − 4·9-s + 49·10-s − 6·11-s + 84·12-s − 7·13-s − 14·14-s − 21·15-s + 210·16-s + 28·18-s − 9·19-s − 196·20-s + 6·21-s + 42·22-s + 7·23-s − 252·24-s + 28·25-s + 49·26-s − 21·27-s + 56·28-s − 4·29-s + ⋯
L(s)  = 1  − 4.94·2-s + 1.73·3-s + 14·4-s − 3.13·5-s − 8.57·6-s + 0.755·7-s − 29.6·8-s − 4/3·9-s + 15.4·10-s − 1.80·11-s + 24.2·12-s − 1.94·13-s − 3.74·14-s − 5.42·15-s + 52.5·16-s + 6.59·18-s − 2.06·19-s − 43.8·20-s + 1.30·21-s + 8.95·22-s + 1.45·23-s − 51.4·24-s + 28/5·25-s + 9.60·26-s − 4.04·27-s + 10.5·28-s − 0.742·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 13^{7} \cdot 31^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 5^{7} \cdot 13^{7} \cdot 31^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(2^{7} \cdot 5^{7} \cdot 13^{7} \cdot 31^{7}\)
Sign: $-1$
Analytic conductor: \(3.57334\times 10^{10}\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(7\)
Selberg data: \((14,\ 2^{7} \cdot 5^{7} \cdot 13^{7} \cdot 31^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{7} \)
5 \( ( 1 + T )^{7} \)
13 \( ( 1 + T )^{7} \)
31 \( ( 1 + T )^{7} \)
good3 \( 1 - p T + 13 T^{2} - 10 p T^{3} + 29 p T^{4} - 55 p T^{5} + 14 p^{3} T^{6} - 608 T^{7} + 14 p^{4} T^{8} - 55 p^{3} T^{9} + 29 p^{4} T^{10} - 10 p^{5} T^{11} + 13 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \)
7 \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 407 T^{4} - 587 T^{5} + 82 p^{2} T^{6} - 4846 T^{7} + 82 p^{3} T^{8} - 587 p^{2} T^{9} + 407 p^{3} T^{10} - 44 p^{4} T^{11} + 27 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 + 6 T + 56 T^{2} + 252 T^{3} + 1368 T^{4} + 4832 T^{5} + 20498 T^{6} + 61264 T^{7} + 20498 p T^{8} + 4832 p^{2} T^{9} + 1368 p^{3} T^{10} + 252 p^{4} T^{11} + 56 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 78 T^{2} - 74 T^{3} + 2809 T^{4} - 4625 T^{5} + 64687 T^{6} - 114612 T^{7} + 64687 p T^{8} - 4625 p^{2} T^{9} + 2809 p^{3} T^{10} - 74 p^{4} T^{11} + 78 p^{5} T^{12} + p^{7} T^{14} \)
19 \( 1 + 9 T + 118 T^{2} + 714 T^{3} + 301 p T^{4} + 27337 T^{5} + 166691 T^{6} + 648560 T^{7} + 166691 p T^{8} + 27337 p^{2} T^{9} + 301 p^{4} T^{10} + 714 p^{4} T^{11} + 118 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 7 T + 101 T^{2} - 720 T^{3} + 5676 T^{4} - 33184 T^{5} + 204025 T^{6} - 937850 T^{7} + 204025 p T^{8} - 33184 p^{2} T^{9} + 5676 p^{3} T^{10} - 720 p^{4} T^{11} + 101 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 + 4 T + 108 T^{2} + 434 T^{3} + 5579 T^{4} + 23923 T^{5} + 199401 T^{6} + 852996 T^{7} + 199401 p T^{8} + 23923 p^{2} T^{9} + 5579 p^{3} T^{10} + 434 p^{4} T^{11} + 108 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 - 2 T + 184 T^{2} - 238 T^{3} + 15638 T^{4} - 12528 T^{5} + 828962 T^{6} - 473498 T^{7} + 828962 p T^{8} - 12528 p^{2} T^{9} + 15638 p^{3} T^{10} - 238 p^{4} T^{11} + 184 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \)
41 \( 1 + 14 T + 278 T^{2} + 2987 T^{3} + 33334 T^{4} + 279879 T^{5} + 2245854 T^{6} + 14837442 T^{7} + 2245854 p T^{8} + 279879 p^{2} T^{9} + 33334 p^{3} T^{10} + 2987 p^{4} T^{11} + 278 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \)
43 \( 1 - 9 T + 204 T^{2} - 1494 T^{3} + 20955 T^{4} - 131267 T^{5} + 1344297 T^{6} - 6970080 T^{7} + 1344297 p T^{8} - 131267 p^{2} T^{9} + 20955 p^{3} T^{10} - 1494 p^{4} T^{11} + 204 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 + 8 T + 231 T^{2} + 1356 T^{3} + 24559 T^{4} + 120745 T^{5} + 1695770 T^{6} + 7043454 T^{7} + 1695770 p T^{8} + 120745 p^{2} T^{9} + 24559 p^{3} T^{10} + 1356 p^{4} T^{11} + 231 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 + 6 T + 176 T^{2} + 213 T^{3} + 9132 T^{4} - 60101 T^{5} + 116572 T^{6} - 5808310 T^{7} + 116572 p T^{8} - 60101 p^{2} T^{9} + 9132 p^{3} T^{10} + 213 p^{4} T^{11} + 176 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 15 T + 175 T^{2} + 1256 T^{3} + 12620 T^{4} + 107168 T^{5} + 1156487 T^{6} + 8645610 T^{7} + 1156487 p T^{8} + 107168 p^{2} T^{9} + 12620 p^{3} T^{10} + 1256 p^{4} T^{11} + 175 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - T + 318 T^{2} - 412 T^{3} + 48194 T^{4} - 63534 T^{5} + 4462834 T^{6} - 5187600 T^{7} + 4462834 p T^{8} - 63534 p^{2} T^{9} + 48194 p^{3} T^{10} - 412 p^{4} T^{11} + 318 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - 14 T + 319 T^{2} - 4186 T^{3} + 52497 T^{4} - 563919 T^{5} + 5455688 T^{6} - 46411550 T^{7} + 5455688 p T^{8} - 563919 p^{2} T^{9} + 52497 p^{3} T^{10} - 4186 p^{4} T^{11} + 319 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 + 16 T + 238 T^{2} + 1272 T^{3} + 8439 T^{4} - 72440 T^{5} - 782715 T^{6} - 13660544 T^{7} - 782715 p T^{8} - 72440 p^{2} T^{9} + 8439 p^{3} T^{10} + 1272 p^{4} T^{11} + 238 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 + 13 T + 554 T^{2} + 5672 T^{3} + 127364 T^{4} + 1032598 T^{5} + 15854580 T^{6} + 100421604 T^{7} + 15854580 p T^{8} + 1032598 p^{2} T^{9} + 127364 p^{3} T^{10} + 5672 p^{4} T^{11} + 554 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + 14 T + 580 T^{2} + 6336 T^{3} + 140796 T^{4} + 1217032 T^{5} + 18725330 T^{6} + 127012532 T^{7} + 18725330 p T^{8} + 1217032 p^{2} T^{9} + 140796 p^{3} T^{10} + 6336 p^{4} T^{11} + 580 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 10 T + 483 T^{2} + 4474 T^{3} + 106427 T^{4} + 871027 T^{5} + 13850882 T^{6} + 94377366 T^{7} + 13850882 p T^{8} + 871027 p^{2} T^{9} + 106427 p^{3} T^{10} + 4474 p^{4} T^{11} + 483 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \)
89 \( 1 + 26 T + 564 T^{2} + 6984 T^{3} + 85831 T^{4} + 762075 T^{5} + 8338319 T^{6} + 69825064 T^{7} + 8338319 p T^{8} + 762075 p^{2} T^{9} + 85831 p^{3} T^{10} + 6984 p^{4} T^{11} + 564 p^{5} T^{12} + 26 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 + 5 T + 153 T^{2} - 1041 T^{3} + 11126 T^{4} - 158237 T^{5} + 1604017 T^{6} - 16566188 T^{7} + 1604017 p T^{8} - 158237 p^{2} T^{9} + 11126 p^{3} T^{10} - 1041 p^{4} T^{11} + 153 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \)
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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

−4.17349755263357163049963968053, −4.16440761531531049921819466947, −3.92282722400180165314815774589, −3.87869773717379800749590954708, −3.34267445453947743695414021753, −3.28890950027427489719765183824, −3.27237468795478968869599500387, −3.24230750396841995580840611253, −3.21429346092917626981287376658, −3.16816884638198704978965352467, −3.12624974124630812403547903126, −2.58887937754258001296456341099, −2.58017237512646593301748413052, −2.52334926385672711163307729923, −2.35312726934202061043221786853, −2.32502645597971016792126122245, −2.32053898314588972209210903095, −2.23545696321459213777794080145, −1.73223223964943059871662304825, −1.59532420836183380004308882251, −1.43749789558430310551429841497, −1.33676136511312899039818848774, −1.32080591339148209740695418648, −0.989885584270938208885598155289, −0.872362291375170782897947843133, 0, 0, 0, 0, 0, 0, 0, 0.872362291375170782897947843133, 0.989885584270938208885598155289, 1.32080591339148209740695418648, 1.33676136511312899039818848774, 1.43749789558430310551429841497, 1.59532420836183380004308882251, 1.73223223964943059871662304825, 2.23545696321459213777794080145, 2.32053898314588972209210903095, 2.32502645597971016792126122245, 2.35312726934202061043221786853, 2.52334926385672711163307729923, 2.58017237512646593301748413052, 2.58887937754258001296456341099, 3.12624974124630812403547903126, 3.16816884638198704978965352467, 3.21429346092917626981287376658, 3.24230750396841995580840611253, 3.27237468795478968869599500387, 3.28890950027427489719765183824, 3.34267445453947743695414021753, 3.87869773717379800749590954708, 3.92282722400180165314815774589, 4.16440761531531049921819466947, 4.17349755263357163049963968053

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

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