Properties

Label 16-13e16-1.1-c3e8-0-0
Degree $16$
Conductor $6.654\times 10^{17}$
Sign $1$
Analytic cond. $9.77287\times 10^{7}$
Root an. cond. $3.15774$
Motivic weight $3$
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  − 10·3-s − 23·4-s + 69·9-s + 230·12-s + 256·16-s + 38·17-s + 392·23-s + 974·25-s − 90·27-s + 88·29-s − 1.58e3·36-s + 574·43-s − 2.56e3·48-s − 303·49-s − 380·51-s − 472·53-s + 2.11e3·61-s − 2.55e3·64-s − 874·68-s − 3.92e3·69-s − 9.74e3·75-s − 4.03e3·79-s − 708·81-s − 880·87-s − 9.01e3·92-s − 2.24e4·100-s + 2.51e3·101-s + ⋯
L(s)  = 1  − 1.92·3-s − 2.87·4-s + 23/9·9-s + 5.53·12-s + 4·16-s + 0.542·17-s + 3.55·23-s + 7.79·25-s − 0.641·27-s + 0.563·29-s − 7.34·36-s + 2.03·43-s − 7.69·48-s − 0.883·49-s − 1.04·51-s − 1.22·53-s + 4.44·61-s − 4.98·64-s − 1.55·68-s − 6.83·69-s − 14.9·75-s − 5.74·79-s − 0.971·81-s − 1.08·87-s − 10.2·92-s − 22.4·100-s + 2.47·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(13^{16}\)
Sign: $1$
Analytic conductor: \(9.77287\times 10^{7}\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{169} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 13^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.259364835\)
\(L(\frac12)\) \(\approx\) \(1.259364835\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + 23 T^{2} + 273 T^{4} + 23 p^{7} T^{6} + 431 p^{6} T^{8} + 23 p^{13} T^{10} + 273 p^{12} T^{12} + 23 p^{18} T^{14} + p^{24} T^{16} \)
3 \( ( 1 + 5 T + p T^{2} - 160 T^{3} - 920 T^{4} - 160 p^{3} T^{5} + p^{7} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
5 \( ( 1 - 487 T^{2} + 90504 T^{4} - 487 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
7 \( 1 + 303 T^{2} - 124787 T^{4} - 5666706 T^{6} + 22271302974 T^{8} - 5666706 p^{6} T^{10} - 124787 p^{12} T^{12} + 303 p^{18} T^{14} + p^{24} T^{16} \)
11 \( 1 + 900 T^{2} + 89198 p T^{4} - 3342870000 T^{6} - 4544090879037 T^{8} - 3342870000 p^{6} T^{10} + 89198 p^{13} T^{12} + 900 p^{18} T^{14} + p^{24} T^{16} \)
17 \( ( 1 - 19 T - 8327 T^{2} + 21622 T^{3} + 49570182 T^{4} + 21622 p^{3} T^{5} - 8327 p^{6} T^{6} - 19 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
19 \( 1 + 15204 T^{2} + 5788798 p T^{4} + 411765249168 T^{6} + 1594731505028259 T^{8} + 411765249168 p^{6} T^{10} + 5788798 p^{13} T^{12} + 15204 p^{18} T^{14} + p^{24} T^{16} \)
23 \( ( 1 - 196 T + 5090 T^{2} - 1762432 T^{3} + 495178915 T^{4} - 1762432 p^{3} T^{5} + 5090 p^{6} T^{6} - 196 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
29 \( ( 1 - 44 T - 8158 T^{2} + 1702096 T^{3} - 540151589 T^{4} + 1702096 p^{3} T^{5} - 8158 p^{6} T^{6} - 44 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
31 \( ( 1 - 105640 T^{2} + 4528623214 T^{4} - 105640 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
37 \( 1 + 180559 T^{2} + 19324352649 T^{4} + 1470787935100826 T^{6} + 86054419198119622214 T^{8} + 1470787935100826 p^{6} T^{10} + 19324352649 p^{12} T^{12} + 180559 p^{18} T^{14} + p^{24} T^{16} \)
41 \( 1 + 245120 T^{2} + 35671467618 T^{4} + 3655263340096000 T^{6} + \)\(28\!\cdots\!43\)\( T^{8} + 3655263340096000 p^{6} T^{10} + 35671467618 p^{12} T^{12} + 245120 p^{18} T^{14} + p^{24} T^{16} \)
43 \( ( 1 - 287 T - 10329 T^{2} + 19032692 T^{3} - 4277355928 T^{4} + 19032692 p^{3} T^{5} - 10329 p^{6} T^{6} - 287 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
47 \( ( 1 - 196231 T^{2} + 19410698292 T^{4} - 196231 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
53 \( ( 1 + 118 T + 297410 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
59 \( 1 + 623108 T^{2} + 215686305018 T^{4} + 54968224538167312 T^{6} + \)\(11\!\cdots\!11\)\( T^{8} + 54968224538167312 p^{6} T^{10} + 215686305018 p^{12} T^{12} + 623108 p^{18} T^{14} + p^{24} T^{16} \)
61 \( ( 1 - 1058 T + 538986 T^{2} - 133748128 T^{3} + 31243888439 T^{4} - 133748128 p^{3} T^{5} + 538986 p^{6} T^{6} - 1058 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
67 \( 1 + 743236 T^{2} + 234440798682 T^{4} + 101854688142795536 T^{6} + \)\(42\!\cdots\!11\)\( T^{8} + 101854688142795536 p^{6} T^{10} + 234440798682 p^{12} T^{12} + 743236 p^{18} T^{14} + p^{24} T^{16} \)
71 \( 1 + 969495 T^{2} + 452631692941 T^{4} + 224038945826147790 T^{6} + \)\(10\!\cdots\!90\)\( T^{8} + 224038945826147790 p^{6} T^{10} + 452631692941 p^{12} T^{12} + 969495 p^{18} T^{14} + p^{24} T^{16} \)
73 \( ( 1 - 877500 T^{2} + 435430002278 T^{4} - 877500 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
79 \( ( 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
83 \( ( 1 - 88712 T^{2} + 116142406846 T^{4} - 88712 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
89 \( 1 + 1281052 T^{2} + 568536628906 T^{4} + 100684302277981552 T^{6} + \)\(87\!\cdots\!39\)\( T^{8} + 100684302277981552 p^{6} T^{10} + 568536628906 p^{12} T^{12} + 1281052 p^{18} T^{14} + p^{24} T^{16} \)
97 \( 1 + 1026556 T^{2} + 68941404042 T^{4} - 699154639691040784 T^{6} - \)\(60\!\cdots\!49\)\( T^{8} - 699154639691040784 p^{6} T^{10} + 68941404042 p^{12} T^{12} + 1026556 p^{18} T^{14} + p^{24} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.37554636507102464001271446753, −5.09733567658340827198782458666, −4.84935027061466530540307304788, −4.74614752038153655446309308091, −4.69771896315871604240149088136, −4.67570963010892306993144185832, −4.48878104232202627639363190747, −4.47711732000732469924892885461, −4.14691272862163332164287113055, −4.02418052105701316264836744354, −3.48442546242046188707169647344, −3.46279201024160671435768254264, −3.22797242206700703142948177038, −2.94073206572822353723126906030, −2.93764608499561134664176727321, −2.83557923392185363812666834261, −2.54061455923627365166880529703, −2.21389689532875192500074892144, −1.38033601424338725115022089696, −1.34090027971695999753591062408, −1.01523314864079918042109118909, −0.962359671806798464011448361724, −0.880282749936896532280897402861, −0.67352219630030157809047347692, −0.19700991366620383009005301668, 0.19700991366620383009005301668, 0.67352219630030157809047347692, 0.880282749936896532280897402861, 0.962359671806798464011448361724, 1.01523314864079918042109118909, 1.34090027971695999753591062408, 1.38033601424338725115022089696, 2.21389689532875192500074892144, 2.54061455923627365166880529703, 2.83557923392185363812666834261, 2.93764608499561134664176727321, 2.94073206572822353723126906030, 3.22797242206700703142948177038, 3.46279201024160671435768254264, 3.48442546242046188707169647344, 4.02418052105701316264836744354, 4.14691272862163332164287113055, 4.47711732000732469924892885461, 4.48878104232202627639363190747, 4.67570963010892306993144185832, 4.69771896315871604240149088136, 4.74614752038153655446309308091, 4.84935027061466530540307304788, 5.09733567658340827198782458666, 5.37554636507102464001271446753

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

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