Properties

Label 16-384e8-1.1-c7e8-0-1
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $4.28717\times 10^{16}$
Root an. cond. $10.9524$
Motivic weight $7$
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  − 2.88e3·7-s − 2.91e3·9-s + 2.28e4·17-s − 2.07e5·23-s + 2.10e5·25-s + 1.78e4·31-s + 6.87e5·41-s + 1.98e6·47-s + 3.26e6·49-s + 8.39e6·63-s + 6.33e6·71-s + 8.92e6·73-s − 1.25e6·79-s + 5.31e6·81-s + 4.44e6·89-s − 1.41e7·97-s + 3.18e7·103-s − 3.59e7·113-s − 6.59e7·119-s + 7.26e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6.67e7·153-s + ⋯
L(s)  = 1  − 3.17·7-s − 4/3·9-s + 1.13·17-s − 3.55·23-s + 2.68·25-s + 0.107·31-s + 1.55·41-s + 2.79·47-s + 3.95·49-s + 4.23·63-s + 2.10·71-s + 2.68·73-s − 0.285·79-s + 10/9·81-s + 0.668·89-s − 1.57·97-s + 2.87·103-s − 2.34·113-s − 3.58·119-s + 3.72·121-s − 1.50·153-s + 11.2·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(4.28717\times 10^{16}\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(19.24760064\)
\(L(\frac12)\) \(\approx\) \(19.24760064\)
\(L(\frac{9}{2})\) 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 \( ( 1 + p^{6} T^{2} )^{4} \)
good5 \( 1 - 210152 T^{2} + 18035115644 T^{4} - 37183502711256 p^{2} T^{6} + 3451278010941686 p^{6} T^{8} - 37183502711256 p^{16} T^{10} + 18035115644 p^{28} T^{12} - 210152 p^{42} T^{14} + p^{56} T^{16} \)
7 \( ( 1 + 1440 T + 30204 p^{2} T^{2} + 1933104672 T^{3} + 2357036209702 T^{4} + 1933104672 p^{7} T^{5} + 30204 p^{16} T^{6} + 1440 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
11 \( 1 - 72660440 T^{2} + 3083374758221948 T^{4} - \)\(91\!\cdots\!36\)\( T^{6} + \)\(20\!\cdots\!90\)\( T^{8} - \)\(91\!\cdots\!36\)\( p^{14} T^{10} + 3083374758221948 p^{28} T^{12} - 72660440 p^{42} T^{14} + p^{56} T^{16} \)
13 \( 1 - 232491496 T^{2} + 32941533643236412 T^{4} - \)\(31\!\cdots\!64\)\( T^{6} + \)\(22\!\cdots\!94\)\( T^{8} - \)\(31\!\cdots\!64\)\( p^{14} T^{10} + 32941533643236412 p^{28} T^{12} - 232491496 p^{42} T^{14} + p^{56} T^{16} \)
17 \( ( 1 - 11448 T + 675462492 T^{2} - 18823419349128 T^{3} + 276096649249682822 T^{4} - 18823419349128 p^{7} T^{5} + 675462492 p^{14} T^{6} - 11448 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
19 \( 1 - 6272279448 T^{2} + 17918977415529239612 T^{4} - \)\(30\!\cdots\!88\)\( T^{6} + \)\(33\!\cdots\!06\)\( T^{8} - \)\(30\!\cdots\!88\)\( p^{14} T^{10} + 17918977415529239612 p^{28} T^{12} - 6272279448 p^{42} T^{14} + p^{56} T^{16} \)
23 \( ( 1 + 103680 T + 9423592732 T^{2} + 713887013378304 T^{3} + 52230530957828619686 T^{4} + 713887013378304 p^{7} T^{5} + 9423592732 p^{14} T^{6} + 103680 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
29 \( 1 - 68451270696 T^{2} + \)\(28\!\cdots\!08\)\( T^{4} - \)\(78\!\cdots\!28\)\( T^{6} + \)\(15\!\cdots\!18\)\( T^{8} - \)\(78\!\cdots\!28\)\( p^{14} T^{10} + \)\(28\!\cdots\!08\)\( p^{28} T^{12} - 68451270696 p^{42} T^{14} + p^{56} T^{16} \)
31 \( ( 1 - 288 p T + 31981062108 T^{2} + 8136257043579552 T^{3} + \)\(19\!\cdots\!42\)\( T^{4} + 8136257043579552 p^{7} T^{5} + 31981062108 p^{14} T^{6} - 288 p^{22} T^{7} + p^{28} T^{8} )^{2} \)
37 \( 1 - 112648835112 T^{2} + \)\(21\!\cdots\!40\)\( T^{4} - \)\(31\!\cdots\!48\)\( T^{6} + \)\(23\!\cdots\!10\)\( T^{8} - \)\(31\!\cdots\!48\)\( p^{14} T^{10} + \)\(21\!\cdots\!40\)\( p^{28} T^{12} - 112648835112 p^{42} T^{14} + p^{56} T^{16} \)
41 \( ( 1 - 343528 T + 365749435452 T^{2} + 35483236220764776 T^{3} + \)\(34\!\cdots\!10\)\( T^{4} + 35483236220764776 p^{7} T^{5} + 365749435452 p^{14} T^{6} - 343528 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
43 \( 1 - 34201218696 p T^{2} + \)\(10\!\cdots\!24\)\( T^{4} - \)\(51\!\cdots\!60\)\( T^{6} + \)\(16\!\cdots\!90\)\( T^{8} - \)\(51\!\cdots\!60\)\( p^{14} T^{10} + \)\(10\!\cdots\!24\)\( p^{28} T^{12} - 34201218696 p^{43} T^{14} + p^{56} T^{16} \)
47 \( ( 1 - 993600 T + 1878320705212 T^{2} - 1254938173705429056 T^{3} + \)\(13\!\cdots\!34\)\( T^{4} - 1254938173705429056 p^{7} T^{5} + 1878320705212 p^{14} T^{6} - 993600 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
53 \( 1 - 4648954961256 T^{2} + \)\(22\!\cdots\!20\)\( p T^{4} - \)\(21\!\cdots\!92\)\( T^{6} + \)\(28\!\cdots\!58\)\( T^{8} - \)\(21\!\cdots\!92\)\( p^{14} T^{10} + \)\(22\!\cdots\!20\)\( p^{29} T^{12} - 4648954961256 p^{42} T^{14} + p^{56} T^{16} \)
59 \( 1 - 9083206385752 T^{2} + \)\(51\!\cdots\!36\)\( T^{4} - \)\(19\!\cdots\!32\)\( T^{6} + \)\(56\!\cdots\!14\)\( T^{8} - \)\(19\!\cdots\!32\)\( p^{14} T^{10} + \)\(51\!\cdots\!36\)\( p^{28} T^{12} - 9083206385752 p^{42} T^{14} + p^{56} T^{16} \)
61 \( 1 - 12317465640296 T^{2} + \)\(81\!\cdots\!36\)\( T^{4} - \)\(38\!\cdots\!68\)\( T^{6} + \)\(13\!\cdots\!98\)\( T^{8} - \)\(38\!\cdots\!68\)\( p^{14} T^{10} + \)\(81\!\cdots\!36\)\( p^{28} T^{12} - 12317465640296 p^{42} T^{14} + p^{56} T^{16} \)
67 \( 1 - 12595380351512 T^{2} + \)\(86\!\cdots\!04\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} - \)\(25\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!00\)\( p^{14} T^{10} + \)\(86\!\cdots\!04\)\( p^{28} T^{12} - 12595380351512 p^{42} T^{14} + p^{56} T^{16} \)
71 \( ( 1 - 3168000 T + 30793182554716 T^{2} - 85783814145563284224 T^{3} + \)\(39\!\cdots\!98\)\( T^{4} - 85783814145563284224 p^{7} T^{5} + 30793182554716 p^{14} T^{6} - 3168000 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
73 \( ( 1 - 4460376 T + 45247419174460 T^{2} - \)\(13\!\cdots\!64\)\( T^{3} + \)\(74\!\cdots\!14\)\( T^{4} - \)\(13\!\cdots\!64\)\( p^{7} T^{5} + 45247419174460 p^{14} T^{6} - 4460376 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
79 \( ( 1 + 625824 T + 24699015020316 T^{2} - 47958880344498060768 T^{3} + \)\(19\!\cdots\!62\)\( T^{4} - 47958880344498060768 p^{7} T^{5} + 24699015020316 p^{14} T^{6} + 625824 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
83 \( 1 - 123395401699736 T^{2} + \)\(80\!\cdots\!28\)\( T^{4} - \)\(35\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!42\)\( T^{8} - \)\(35\!\cdots\!72\)\( p^{14} T^{10} + \)\(80\!\cdots\!28\)\( p^{28} T^{12} - 123395401699736 p^{42} T^{14} + p^{56} T^{16} \)
89 \( ( 1 - 2223704 T + 82674152988156 T^{2} - 35737168093538692776 T^{3} + \)\(40\!\cdots\!18\)\( T^{4} - 35737168093538692776 p^{7} T^{5} + 82674152988156 p^{14} T^{6} - 2223704 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
97 \( ( 1 + 7078792 T + 170684496494492 T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} + \)\(18\!\cdots\!24\)\( p^{7} T^{5} + 170684496494492 p^{14} T^{6} + 7078792 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
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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

−3.67559682283970249798378858756, −3.63104084246193768868545770378, −3.37885786435595855973513588789, −3.34477562605931961853015176298, −3.02220803806875801174290096343, −3.01703858368843878614067710983, −2.97938748434043107269759535960, −2.92366085247958399269290819395, −2.74791146450889104725776307494, −2.72142100176581559890178829571, −2.21254250871377383376030570992, −2.06299438034111372290099995320, −2.05269452421564668918993536401, −1.99000556502441482956933809263, −1.69233625513371074230877807454, −1.63867205317007382070271855141, −1.61351156090278522720830235838, −0.872899795729505294207254685799, −0.75963661897028892099466071207, −0.73458187609075718150336185087, −0.70139822949301773148035939756, −0.60294343758254603802891179520, −0.50389771009693327797449112117, −0.42397603194673744461725598533, −0.21317241365070083733338121307, 0.21317241365070083733338121307, 0.42397603194673744461725598533, 0.50389771009693327797449112117, 0.60294343758254603802891179520, 0.70139822949301773148035939756, 0.73458187609075718150336185087, 0.75963661897028892099466071207, 0.872899795729505294207254685799, 1.61351156090278522720830235838, 1.63867205317007382070271855141, 1.69233625513371074230877807454, 1.99000556502441482956933809263, 2.05269452421564668918993536401, 2.06299438034111372290099995320, 2.21254250871377383376030570992, 2.72142100176581559890178829571, 2.74791146450889104725776307494, 2.92366085247958399269290819395, 2.97938748434043107269759535960, 3.01703858368843878614067710983, 3.02220803806875801174290096343, 3.34477562605931961853015176298, 3.37885786435595855973513588789, 3.63104084246193768868545770378, 3.67559682283970249798378858756

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

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