Properties

Label 16-12e24-1.1-c2e8-0-15
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
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  + 100·25-s − 276·41-s + 196·49-s + 568·73-s + 188·97-s + 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 4·25-s − 6.73·41-s + 4·49-s + 7.78·73-s + 1.93·97-s + 0.380·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(18.92038661\)
\(L(\frac12)\) \(\approx\) \(18.92038661\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \)
7 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \)
11 \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2}( 1 + 46 T^{2} - 12525 T^{4} + 46 p^{4} T^{6} + p^{8} T^{8} ) \)
13 \( ( 1 - p T + p^{2} T^{2} )^{4}( 1 + p T + p^{2} T^{2} )^{4} \)
17 \( ( 1 - 2 T - 285 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2}( 1 + 2 T - 285 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19 \( ( 1 - 434 T^{2} + 58035 T^{4} - 434 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 - p T + p^{2} T^{2} )^{4}( 1 + p T + p^{2} T^{2} )^{4} \)
29 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \)
31 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
41 \( ( 1 + 46 T + p^{2} T^{2} )^{4}( 1 + 46 T + 435 T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43 \( ( 1 - 3502 T^{2} + p^{4} T^{4} )^{2}( 1 + 3502 T^{2} + 8845203 T^{4} + 3502 p^{4} T^{6} + p^{8} T^{8} ) \)
47 \( ( 1 - p T + p^{2} T^{2} )^{4}( 1 + p T + p^{2} T^{2} )^{4} \)
53 \( ( 1 + p^{2} T^{2} )^{8} \)
59 \( ( 1 - 238 T^{2} + p^{4} T^{4} )^{2}( 1 + 238 T^{2} - 12060717 T^{4} + 238 p^{4} T^{6} + p^{8} T^{8} ) \)
61 \( ( 1 - p T + p^{2} T^{2} )^{4}( 1 + p T + p^{2} T^{2} )^{4} \)
67 \( ( 1 - 5134 T^{2} + p^{4} T^{4} )^{2}( 1 + 5134 T^{2} + 6206835 T^{4} + 5134 p^{4} T^{6} + p^{8} T^{8} ) \)
71 \( ( 1 - p T )^{8}( 1 + p T )^{8} \)
73 \( ( 1 - 142 T + 14835 T^{2} - 142 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
79 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \)
83 \( ( 1 - 11186 T^{2} + 77668275 T^{4} - 11186 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 146 T + p^{2} T^{2} )^{4}( 1 + 146 T + p^{2} T^{2} )^{4} \)
97 \( ( 1 - 94 T + p^{2} T^{2} )^{4}( 1 + 94 T - 573 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−3.65651744079417453919041851890, −3.55401210688043083079547678737, −3.47719650511476479303333980988, −3.32480402560909646260387120623, −3.30351520695225561221786305070, −3.06189718354813852856501766917, −3.01701910521450791838472218150, −2.85989245868026909232503599723, −2.83841974900349880736004265829, −2.46413791391973787909324452100, −2.26189058938546931961436107111, −2.25877241382465167900887051558, −2.19398265300968441305522699961, −2.08566744565547507739634772262, −1.93601606051222720356938250176, −1.63585024254734175769364888712, −1.59212778507203244666791453178, −1.33589480204482673601146730116, −1.26231005021808722571082185398, −0.971716200136228624585365120800, −0.75832165683613007109042233256, −0.67373127773753113255485077675, −0.58576844388520280887617763059, −0.46751496546896168731086728076, −0.21844186677089650518409618045, 0.21844186677089650518409618045, 0.46751496546896168731086728076, 0.58576844388520280887617763059, 0.67373127773753113255485077675, 0.75832165683613007109042233256, 0.971716200136228624585365120800, 1.26231005021808722571082185398, 1.33589480204482673601146730116, 1.59212778507203244666791453178, 1.63585024254734175769364888712, 1.93601606051222720356938250176, 2.08566744565547507739634772262, 2.19398265300968441305522699961, 2.25877241382465167900887051558, 2.26189058938546931961436107111, 2.46413791391973787909324452100, 2.83841974900349880736004265829, 2.85989245868026909232503599723, 3.01701910521450791838472218150, 3.06189718354813852856501766917, 3.30351520695225561221786305070, 3.32480402560909646260387120623, 3.47719650511476479303333980988, 3.55401210688043083079547678737, 3.65651744079417453919041851890

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

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