Properties

Label 16-171e8-1.1-c2e8-0-0
Degree $16$
Conductor $7.311\times 10^{17}$
Sign $1$
Analytic cond. $222151.$
Root an. cond. $2.15856$
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  − 2·4-s − 40·7-s − 60·13-s + 3·16-s − 96·19-s + 32·25-s + 80·28-s − 100·43-s + 628·49-s + 120·52-s − 68·61-s + 54·64-s − 180·67-s − 60·73-s + 192·76-s + 420·79-s + 2.40e3·91-s − 840·97-s − 64·100-s − 648·109-s − 120·112-s − 688·121-s + 127-s + 131-s + 3.84e3·133-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 5.71·7-s − 4.61·13-s + 3/16·16-s − 5.05·19-s + 1.27·25-s + 20/7·28-s − 2.32·43-s + 12.8·49-s + 2.30·52-s − 1.11·61-s + 0.843·64-s − 2.68·67-s − 0.821·73-s + 2.52·76-s + 5.31·79-s + 26.3·91-s − 8.65·97-s − 0.639·100-s − 5.94·109-s − 1.07·112-s − 5.68·121-s + 0.00787·127-s + 0.00763·131-s + 28.8·133-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(222151.\)
Root analytic conductor: \(2.15856\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{171} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 19^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.02605332510\)
\(L(\frac12)\) \(\approx\) \(0.02605332510\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( ( 1 + 48 T + 62 p T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
good2 \( 1 + p T^{2} + T^{4} - 29 p T^{6} - 311 T^{8} - 29 p^{5} T^{10} + p^{8} T^{12} + p^{13} T^{14} + p^{16} T^{16} \)
5 \( 1 - 32 T^{2} + 598 T^{4} + 26368 T^{6} - 849149 T^{8} + 26368 p^{4} T^{10} + 598 p^{8} T^{12} - 32 p^{12} T^{14} + p^{16} T^{16} \)
7 \( ( 1 + 10 T + 93 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
11 \( ( 1 + 344 T^{2} + 55866 T^{4} + 344 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 30 T + 673 T^{2} + 11190 T^{3} + 161268 T^{4} + 11190 p^{2} T^{5} + 673 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
17 \( 1 - 988 T^{2} + 569386 T^{4} - 236839408 T^{6} + 76947790099 T^{8} - 236839408 p^{4} T^{10} + 569386 p^{8} T^{12} - 988 p^{12} T^{14} + p^{16} T^{16} \)
23 \( 1 + 1216 T^{2} + 552310 T^{4} + 445863424 T^{6} + 366017352739 T^{8} + 445863424 p^{4} T^{10} + 552310 p^{8} T^{12} + 1216 p^{12} T^{14} + p^{16} T^{16} \)
29 \( 1 + 1964 T^{2} + 1778410 T^{4} + 1304732336 T^{6} + 1093328415859 T^{8} + 1304732336 p^{4} T^{10} + 1778410 p^{8} T^{12} + 1964 p^{12} T^{14} + p^{16} T^{16} \)
31 \( ( 1 - 3098 T^{2} + 4231923 T^{4} - 3098 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 1006 T^{2} + 1301331 T^{4} - 1006 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
41 \( 1 + 4004 T^{2} + 8100490 T^{4} + 9129136016 T^{6} + 9621533895379 T^{8} + 9129136016 p^{4} T^{10} + 8100490 p^{8} T^{12} + 4004 p^{12} T^{14} + p^{16} T^{16} \)
43 \( ( 1 + 50 T - 353 T^{2} - 42250 T^{3} + 98308 T^{4} - 42250 p^{2} T^{5} - 353 p^{4} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
47 \( 1 - 4868 T^{2} + 8136586 T^{4} - 28241585168 T^{6} + 102692522958739 T^{8} - 28241585168 p^{4} T^{10} + 8136586 p^{8} T^{12} - 4868 p^{12} T^{14} + p^{16} T^{16} \)
53 \( 1 + 4192 T^{2} + 4138966 T^{4} - 9838892288 T^{6} - 20448424444541 T^{8} - 9838892288 p^{4} T^{10} + 4138966 p^{8} T^{12} + 4192 p^{12} T^{14} + p^{16} T^{16} \)
59 \( 1 - 1576 T^{2} - 20496890 T^{4} + 1976392256 T^{6} + 351183761764099 T^{8} + 1976392256 p^{4} T^{10} - 20496890 p^{8} T^{12} - 1576 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 34 T - 3575 T^{2} - 92174 T^{3} + 4235044 T^{4} - 92174 p^{2} T^{5} - 3575 p^{4} T^{6} + 34 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 90 T + 11143 T^{2} + 759870 T^{3} + 63253428 T^{4} + 759870 p^{2} T^{5} + 11143 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
71 \( 1 + 9484 T^{2} + 25384330 T^{4} + 130296540976 T^{6} + 1236803192967379 T^{8} + 130296540976 p^{4} T^{10} + 25384330 p^{8} T^{12} + 9484 p^{12} T^{14} + p^{16} T^{16} \)
73 \( ( 1 + 30 T - 263 T^{2} - 284850 T^{3} - 31841772 T^{4} - 284850 p^{2} T^{5} - 263 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
79 \( ( 1 - 210 T + 27247 T^{2} - 2634870 T^{3} + 210219828 T^{4} - 2634870 p^{2} T^{5} + 27247 p^{4} T^{6} - 210 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( ( 1 + 20348 T^{2} + 197965638 T^{4} + 20348 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( 1 - 896 T^{2} - 70207370 T^{4} + 48808969216 T^{6} + 1055487235101859 T^{8} + 48808969216 p^{4} T^{10} - 70207370 p^{8} T^{12} - 896 p^{12} T^{14} + p^{16} T^{16} \)
97 \( ( 1 + 420 T + 91678 T^{2} + 13808760 T^{3} + 1545682803 T^{4} + 13808760 p^{2} T^{5} + 91678 p^{4} T^{6} + 420 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
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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

−5.60369901948174533915352557726, −5.47694688333101171389857247221, −5.06473101769695880784597108832, −5.05336319159583949890029035533, −5.02220131172903057413462541666, −5.02183892025749073444846638835, −4.33581256701005924747688623182, −4.26571464315779009862996505949, −4.19481637671908257042329553979, −4.17583081577912541164362512876, −3.98120492427331931168961268045, −3.79778835146608226904934236713, −3.28428691420936871192945215184, −3.09350933367236979500708433694, −3.04390036225751004669657501590, −2.96145746912870371067596575831, −2.73491459731614340969944701637, −2.72130761790985842943513473213, −2.42019570254218475076922660464, −2.05800577660206821932560172786, −1.73756186639596990540366529544, −1.67087352388340331862219089436, −0.40721440269600467823344194697, −0.24189237291714546427302848096, −0.16441367624798546082912033082, 0.16441367624798546082912033082, 0.24189237291714546427302848096, 0.40721440269600467823344194697, 1.67087352388340331862219089436, 1.73756186639596990540366529544, 2.05800577660206821932560172786, 2.42019570254218475076922660464, 2.72130761790985842943513473213, 2.73491459731614340969944701637, 2.96145746912870371067596575831, 3.04390036225751004669657501590, 3.09350933367236979500708433694, 3.28428691420936871192945215184, 3.79778835146608226904934236713, 3.98120492427331931168961268045, 4.17583081577912541164362512876, 4.19481637671908257042329553979, 4.26571464315779009862996505949, 4.33581256701005924747688623182, 5.02183892025749073444846638835, 5.02220131172903057413462541666, 5.05336319159583949890029035533, 5.06473101769695880784597108832, 5.47694688333101171389857247221, 5.60369901948174533915352557726

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

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