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Properties

Label 16-39e8-1.1-c2e8-0-1
Degree $16$
Conductor $5.352\times 10^{12}$
Sign $1$
Analytic cond. $1.62628$
Root an. cond. $1.03086$
Motivic weight $2$
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 + 6·4-s + 8·7-s − 9-s − 12·12-s + 4·16-s − 8·19-s − 16·21-s + 70·25-s − 10·27-s + 48·28-s + 80·31-s − 6·36-s − 32·37-s − 196·43-s − 8·48-s − 174·49-s + 16·57-s + 248·61-s − 8·63-s + 10·64-s − 176·67-s + 440·73-s − 140·75-s − 48·76-s − 88·79-s − 63·81-s + ⋯
L(s)  = 1  − 2/3·3-s + 3/2·4-s + 8/7·7-s − 1/9·9-s − 12-s + 1/4·16-s − 0.421·19-s − 0.761·21-s + 14/5·25-s − 0.370·27-s + 12/7·28-s + 2.58·31-s − 1/6·36-s − 0.864·37-s − 4.55·43-s − 1/6·48-s − 3.55·49-s + 0.280·57-s + 4.06·61-s − 0.126·63-s + 5/32·64-s − 2.62·67-s + 6.02·73-s − 1.86·75-s − 0.631·76-s − 1.11·79-s − 7/9·81-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(1.62628\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 13^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.521338172\)
\(L(\frac12)\) \(\approx\) \(1.521338172\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 2 T + 5 T^{2} + 22 T^{3} + 44 p T^{4} + 22 p^{2} T^{5} + 5 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \)
13 \( ( 1 - p T^{2} )^{4} \)
good2 \( 1 - 3 p T^{2} + p^{5} T^{4} - 89 p T^{6} + 695 T^{8} - 89 p^{5} T^{10} + p^{13} T^{12} - 3 p^{13} T^{14} + p^{16} T^{16} \)
5 \( 1 - 14 p T^{2} + 549 p T^{4} - 85562 T^{6} + 2303644 T^{8} - 85562 p^{4} T^{10} + 549 p^{9} T^{12} - 14 p^{13} T^{14} + p^{16} T^{16} \)
7 \( ( 1 - 4 T + 111 T^{2} - 124 T^{3} + 5648 T^{4} - 124 p^{2} T^{5} + 111 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
11 \( 1 - 500 T^{2} + 98260 T^{4} - 9861900 T^{6} + 852333462 T^{8} - 9861900 p^{4} T^{10} + 98260 p^{8} T^{12} - 500 p^{12} T^{14} + p^{16} T^{16} \)
17 \( 1 - 1506 T^{2} + 1089041 T^{4} - 504329362 T^{6} + 168439474244 T^{8} - 504329362 p^{4} T^{10} + 1089041 p^{8} T^{12} - 1506 p^{12} T^{14} + p^{16} T^{16} \)
19 \( ( 1 + 4 T + 1268 T^{2} + 3868 T^{3} + 658742 T^{4} + 3868 p^{2} T^{5} + 1268 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
23 \( 1 - 80 p T^{2} + 1414172 T^{4} - 24850608 p T^{6} + 205265712710 T^{8} - 24850608 p^{5} T^{10} + 1414172 p^{8} T^{12} - 80 p^{13} T^{14} + p^{16} T^{16} \)
29 \( 1 - 4232 T^{2} + 9016156 T^{4} - 12505941816 T^{6} + 12341247010566 T^{8} - 12505941816 p^{4} T^{10} + 9016156 p^{8} T^{12} - 4232 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - 40 T + 2156 T^{2} - 70232 T^{3} + 2314278 T^{4} - 70232 p^{2} T^{5} + 2156 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 + 16 T + 3011 T^{2} + 47456 T^{3} + 6010968 T^{4} + 47456 p^{2} T^{5} + 3011 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
41 \( 1 - 172 p T^{2} + 26262164 T^{4} - 66905143860 T^{6} + 127601566849750 T^{8} - 66905143860 p^{4} T^{10} + 26262164 p^{8} T^{12} - 172 p^{13} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 98 T + 9925 T^{2} + 557210 T^{3} + 29823852 T^{4} + 557210 p^{2} T^{5} + 9925 p^{4} T^{6} + 98 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 13902 T^{2} + 91655345 T^{4} - 369070610034 T^{6} + 988644664659852 T^{8} - 369070610034 p^{4} T^{10} + 91655345 p^{8} T^{12} - 13902 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 + 512 T^{2} + 25800764 T^{4} + 8660148480 T^{6} + 282138540156998 T^{8} + 8660148480 p^{4} T^{10} + 25800764 p^{8} T^{12} + 512 p^{12} T^{14} + p^{16} T^{16} \)
59 \( 1 - 3044 T^{2} + 18737204 T^{4} - 24634623516 T^{6} + 226280659565014 T^{8} - 24634623516 p^{4} T^{10} + 18737204 p^{8} T^{12} - 3044 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 - 124 T + 20468 T^{2} - 1492196 T^{3} + 125387190 T^{4} - 1492196 p^{2} T^{5} + 20468 p^{4} T^{6} - 124 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 88 T + 16076 T^{2} + 998888 T^{3} + 106376454 T^{4} + 998888 p^{2} T^{5} + 16076 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( 1 - 19918 T^{2} + 184236785 T^{4} - 1070790254322 T^{6} + 5313378473821004 T^{8} - 1070790254322 p^{4} T^{10} + 184236785 p^{8} T^{12} - 19918 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 - 220 T + 34344 T^{2} - 3496084 T^{3} + 296170958 T^{4} - 3496084 p^{2} T^{5} + 34344 p^{4} T^{6} - 220 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( ( 1 + 44 T + 5280 T^{2} + 553172 T^{3} + 41572094 T^{4} + 553172 p^{2} T^{5} + 5280 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( 1 - 29268 T^{2} + 483126932 T^{4} - 5320275437356 T^{6} + 42845471887251350 T^{8} - 5320275437356 p^{4} T^{10} + 483126932 p^{8} T^{12} - 29268 p^{12} T^{14} + p^{16} T^{16} \)
89 \( 1 - 41164 T^{2} + 863014548 T^{4} - 11659840670708 T^{6} + 109502594368397782 T^{8} - 11659840670708 p^{4} T^{10} + 863014548 p^{8} T^{12} - 41164 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 + 160 T + 36680 T^{2} + 3724208 T^{3} + 493608654 T^{4} + 3724208 p^{2} T^{5} + 36680 p^{4} T^{6} + 160 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−7.77586253225680561479835067131, −7.65502613675886366932298340134, −7.19142472366556181486893050489, −6.95026158346670135754300451805, −6.77731904571882783091168861212, −6.77291195943969582978961284996, −6.69113981577489505505711560555, −6.54256015833065443295000909480, −6.25571392047480323840862051512, −6.08280045909784638197987960554, −5.64664812918473059153005463295, −5.31862082496627999757999874058, −5.19716777718374009485411788275, −5.06259849347282237239985345271, −4.88448643323950613143255652227, −4.69374326495256836340639132179, −4.25989403485480194919266416193, −4.14149368153317084281866599385, −3.46249926164402682189529456198, −3.30494954952340122213412262607, −3.05961819905227757891147826314, −2.68074860430408997753465912209, −2.11778574800441757324209283040, −1.90585497737636563986785070600, −1.24925668789915053436805909303, 1.24925668789915053436805909303, 1.90585497737636563986785070600, 2.11778574800441757324209283040, 2.68074860430408997753465912209, 3.05961819905227757891147826314, 3.30494954952340122213412262607, 3.46249926164402682189529456198, 4.14149368153317084281866599385, 4.25989403485480194919266416193, 4.69374326495256836340639132179, 4.88448643323950613143255652227, 5.06259849347282237239985345271, 5.19716777718374009485411788275, 5.31862082496627999757999874058, 5.64664812918473059153005463295, 6.08280045909784638197987960554, 6.25571392047480323840862051512, 6.54256015833065443295000909480, 6.69113981577489505505711560555, 6.77291195943969582978961284996, 6.77731904571882783091168861212, 6.95026158346670135754300451805, 7.19142472366556181486893050489, 7.65502613675886366932298340134, 7.77586253225680561479835067131

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

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