Properties

Label 16-1216e8-1.1-c1e8-0-4
Degree $16$
Conductor $4.780\times 10^{24}$
Sign $1$
Analytic cond. $7.90106\times 10^{7}$
Root an. cond. $3.11605$
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  + 2·9-s + 24·17-s − 20·25-s − 12·41-s − 56·49-s + 4·73-s − 5·81-s − 72·89-s + 20·97-s − 72·113-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s + 5.82·17-s − 4·25-s − 1.87·41-s − 8·49-s + 0.468·73-s − 5/9·81-s − 7.63·89-s + 2.03·97-s − 6.77·113-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(7.90106\times 10^{7}\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.122425497\)
\(L(\frac12)\) \(\approx\) \(1.122425497\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \)
good3 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2}( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} ) \)
5 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
7 \( ( 1 + p T^{2} )^{8} \)
11 \( ( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + p T^{2} )^{8} \)
37 \( ( 1 - p T^{2} )^{8} \)
41 \( ( 1 + 6 T + p T^{2} )^{4}( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43 \( ( 1 - 14 T^{2} - 1653 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2}( 1 + 82 T^{2} + 3243 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} ) \)
61 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} ) \)
71 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 - 2 T + p T^{2} )^{4}( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
79 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−4.17524421400586638924160037825, −3.94591446448797241677003699211, −3.85999469430684076935262763719, −3.61968852446160698472193884090, −3.59230494809347552591647802219, −3.55259535391697080685741596785, −3.36184773178340084390222758645, −3.30329491063332326092019091470, −3.09597050278135129430632393828, −3.06480818071681350355869200140, −2.94030145689785725046673071303, −2.67580474157692121101549556324, −2.62608157289118804858292920330, −2.44751952398668393847602128250, −2.19177559420183175303914705057, −1.83687624291780246376350889663, −1.79805201381189175803210083353, −1.58115220263223997768351981511, −1.48659243628697906503626100352, −1.43990695991391989017928792395, −1.35458198694114020676825062594, −1.22129736066798647178126458339, −0.792564002647747263379544991258, −0.30989240144337206409754988338, −0.16169275783744718460747749585, 0.16169275783744718460747749585, 0.30989240144337206409754988338, 0.792564002647747263379544991258, 1.22129736066798647178126458339, 1.35458198694114020676825062594, 1.43990695991391989017928792395, 1.48659243628697906503626100352, 1.58115220263223997768351981511, 1.79805201381189175803210083353, 1.83687624291780246376350889663, 2.19177559420183175303914705057, 2.44751952398668393847602128250, 2.62608157289118804858292920330, 2.67580474157692121101549556324, 2.94030145689785725046673071303, 3.06480818071681350355869200140, 3.09597050278135129430632393828, 3.30329491063332326092019091470, 3.36184773178340084390222758645, 3.55259535391697080685741596785, 3.59230494809347552591647802219, 3.61968852446160698472193884090, 3.85999469430684076935262763719, 3.94591446448797241677003699211, 4.17524421400586638924160037825

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

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