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Properties

Label 16-384e8-1.1-c1e8-0-0
Degree $16$
Conductor $4.728\times 10^{20}$
Sign $1$
Analytic cond. $7813.87$
Root an. cond. $1.75107$
Motivic weight $1$
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  − 8·11-s − 8·19-s + 16·29-s − 24·31-s + 16·37-s − 8·43-s + 24·49-s − 16·53-s + 32·59-s − 16·61-s − 16·67-s + 24·79-s − 2·81-s − 40·83-s − 32·107-s + 16·113-s + 32·121-s − 16·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2.41·11-s − 1.83·19-s + 2.97·29-s − 4.31·31-s + 2.63·37-s − 1.21·43-s + 24/7·49-s − 2.19·53-s + 4.16·59-s − 2.04·61-s − 1.95·67-s + 2.70·79-s − 2/9·81-s − 4.39·83-s − 3.09·107-s + 1.50·113-s + 2.90·121-s − 1.43·125-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(7813.87\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.026629876\)
\(L(\frac12)\) \(\approx\) \(1.026629876\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{4} )^{2} \)
good5 \( 1 + 16 T^{3} - 12 T^{4} - 48 T^{5} + 128 T^{6} + 32 T^{7} - 506 T^{8} + 32 p T^{9} + 128 p^{2} T^{10} - 48 p^{3} T^{11} - 12 p^{4} T^{12} + 16 p^{5} T^{13} + p^{8} T^{16} \)
7 \( 1 - 24 T^{2} + 292 T^{4} - 2440 T^{6} + 17222 T^{8} - 2440 p^{2} T^{10} + 292 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + 12 p T^{4} + 344 T^{5} + 2400 T^{6} + 13000 T^{7} + 54374 T^{8} + 13000 p T^{9} + 2400 p^{2} T^{10} + 344 p^{3} T^{11} + 12 p^{5} T^{12} + 8 p^{6} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + 64 T^{3} - 4 T^{4} - 704 T^{5} + 2048 T^{6} + 1408 T^{7} - 53466 T^{8} + 1408 p T^{9} + 2048 p^{2} T^{10} - 704 p^{3} T^{11} - 4 p^{4} T^{12} + 64 p^{5} T^{13} + p^{8} T^{16} \)
17 \( ( 1 + 36 T^{2} + 64 T^{3} + 662 T^{4} + 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 452 T^{4} + 2168 T^{5} + 10080 T^{6} + 37832 T^{7} + 138918 T^{8} + 37832 p T^{9} + 10080 p^{2} T^{10} + 2168 p^{3} T^{11} + 452 p^{4} T^{12} + 120 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
23 \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \)
29 \( 1 - 16 T + 128 T^{2} - 32 p T^{3} + 6580 T^{4} - 38208 T^{5} + 199680 T^{6} - 1073680 T^{7} + 5802054 T^{8} - 1073680 p T^{9} + 199680 p^{2} T^{10} - 38208 p^{3} T^{11} + 6580 p^{4} T^{12} - 32 p^{6} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
31 \( ( 1 + 12 T + 164 T^{2} + 1140 T^{3} + 8218 T^{4} + 1140 p T^{5} + 164 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 16 T + 128 T^{2} - 1008 T^{3} + 5948 T^{4} - 15248 T^{5} - 9344 T^{6} + 717840 T^{7} - 7530650 T^{8} + 717840 p T^{9} - 9344 p^{2} T^{10} - 15248 p^{3} T^{11} + 5948 p^{4} T^{12} - 1008 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 200 T^{2} + 19452 T^{4} - 1244536 T^{6} + 58583750 T^{8} - 1244536 p^{2} T^{10} + 19452 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 + 8 T + 32 T^{2} + 56 T^{3} + 260 T^{4} + 504 T^{5} - 2720 T^{6} - 625528 T^{7} - 7635866 T^{8} - 625528 p T^{9} - 2720 p^{2} T^{10} + 504 p^{3} T^{11} + 260 p^{4} T^{12} + 56 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
47 \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \)
53 \( 1 + 16 T + 128 T^{2} + 928 T^{3} + 8564 T^{4} + 82496 T^{5} + 654336 T^{6} + 5021328 T^{7} + 38116486 T^{8} + 5021328 p T^{9} + 654336 p^{2} T^{10} + 82496 p^{3} T^{11} + 8564 p^{4} T^{12} + 928 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
61 \( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 14204 T^{4} + 79760 T^{5} + 426880 T^{6} + 2945904 T^{7} + 19569574 T^{8} + 2945904 p T^{9} + 426880 p^{2} T^{10} + 79760 p^{3} T^{11} + 14204 p^{4} T^{12} + 1392 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 16 T + 128 T^{2} + 304 T^{3} + 4388 T^{4} + 107696 T^{5} + 1207680 T^{6} + 4800272 T^{7} + 13154790 T^{8} + 4800272 p T^{9} + 1207680 p^{2} T^{10} + 107696 p^{3} T^{11} + 4388 p^{4} T^{12} + 304 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 440 T^{2} + 90844 T^{4} - 11522952 T^{6} + 984512390 T^{8} - 11522952 p^{2} T^{10} + 90844 p^{4} T^{12} - 440 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 328 T^{2} + 45404 T^{4} - 3734648 T^{6} + 259745542 T^{8} - 3734648 p^{2} T^{10} + 45404 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 12 T + 148 T^{2} + 44 T^{3} + 794 T^{4} + 44 p T^{5} + 148 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 40 T + 800 T^{2} + 11000 T^{3} + 122436 T^{4} + 1297720 T^{5} + 14460000 T^{6} + 161033000 T^{7} + 1597489574 T^{8} + 161033000 p T^{9} + 14460000 p^{2} T^{10} + 1297720 p^{3} T^{11} + 122436 p^{4} T^{12} + 11000 p^{5} T^{13} + 800 p^{6} T^{14} + 40 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 248 T^{2} + 36316 T^{4} - 4626504 T^{6} + 476004998 T^{8} - 4626504 p^{2} T^{10} + 36316 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−5.06580784456107007101824914050, −4.94224327695178115308891158360, −4.71702490166394994132926119716, −4.57303891753490048489946055690, −4.45080005763411113761176764510, −4.24033487173737568307061844510, −4.19294989693500986486603577566, −4.03094278598254391732269643485, −3.91344490704705095475928625011, −3.86161523887856243358622988348, −3.55113341355270170593859663200, −3.09915399267279008214102448632, −3.06489859632359584099818803741, −2.95247376561598072688580614664, −2.87644921536400208707403928719, −2.83021223977478145446916601254, −2.50777444348019446356960492880, −2.16088660777696605820702364378, −2.15926500318119837396991651153, −1.81941199671781103511855085726, −1.76995859892079450604564584527, −1.57892332046813321115411444929, −0.837226976059706259047394949791, −0.798323976232333238655454651688, −0.26117879079963944313896189029, 0.26117879079963944313896189029, 0.798323976232333238655454651688, 0.837226976059706259047394949791, 1.57892332046813321115411444929, 1.76995859892079450604564584527, 1.81941199671781103511855085726, 2.15926500318119837396991651153, 2.16088660777696605820702364378, 2.50777444348019446356960492880, 2.83021223977478145446916601254, 2.87644921536400208707403928719, 2.95247376561598072688580614664, 3.06489859632359584099818803741, 3.09915399267279008214102448632, 3.55113341355270170593859663200, 3.86161523887856243358622988348, 3.91344490704705095475928625011, 4.03094278598254391732269643485, 4.19294989693500986486603577566, 4.24033487173737568307061844510, 4.45080005763411113761176764510, 4.57303891753490048489946055690, 4.71702490166394994132926119716, 4.94224327695178115308891158360, 5.06580784456107007101824914050

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

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