Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·11-s + 16-s + 36·29-s + 20·31-s − 12·41-s − 36·49-s − 36·59-s + 16·61-s − 18·81-s − 12·101-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 24·176-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 7.23·11-s + 1/4·16-s + 6.68·29-s + 3.59·31-s − 1.87·41-s − 5.14·49-s − 4.68·59-s + 2.04·61-s − 2·81-s − 1.19·101-s + 24.3·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.76·169-s + 0.0760·173-s + 1.80·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{450} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(13.0595\)
\(L(\frac12)\)  \(\approx\)  \(13.0595\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( ( 1 + p^{2} T^{4} )^{2} \)
5 \( 1 \)
good7 \( 1 + 36 T^{2} + 634 T^{4} + 7272 T^{6} + 59571 T^{8} + 7272 p^{2} T^{10} + 634 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
13 \( 1 + 36 T^{2} + 682 T^{4} + 9000 T^{6} + 106947 T^{8} + 9000 p^{2} T^{10} + 682 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 14 T^{2} + 339 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( 1 + 108 T^{2} + 5818 T^{4} + 208440 T^{6} + 5501811 T^{8} + 208440 p^{2} T^{10} + 5818 p^{4} T^{12} + 108 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 18 T + 188 T^{2} - 1404 T^{3} + 8259 T^{4} - 1404 p T^{5} + 188 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 10 T + 40 T^{2} + 20 T^{3} - 461 T^{4} + 20 p T^{5} + 40 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 644 T^{4} - 1762266 T^{8} - 644 p^{4} T^{12} + p^{8} T^{16} \)
41 \( ( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \)
43 \( 1 + 72 T^{2} + 1633 T^{4} - 6840 T^{6} - 214704 T^{8} - 6840 p^{2} T^{10} + 1633 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 1054 T^{4} - 3768765 T^{8} + 1054 p^{4} T^{12} + p^{8} T^{16} \)
53 \( 1 + 508 T^{4} - 3205722 T^{8} + 508 p^{4} T^{12} + p^{8} T^{16} \)
59 \( ( 1 + 18 T + 137 T^{2} + 1242 T^{3} + 12372 T^{4} + 1242 p T^{5} + 137 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( 1 + 216 T^{2} + 26737 T^{4} + 2415960 T^{6} + 174766032 T^{8} + 2415960 p^{2} T^{10} + 26737 p^{4} T^{12} + 216 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 116 T^{2} + 12474 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 3503 T^{4} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 58 T^{2} - 2877 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( 1 + 432 T^{2} + 91513 T^{4} + 12659760 T^{6} + 1239875616 T^{8} + 12659760 p^{2} T^{10} + 91513 p^{4} T^{12} + 432 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 103 T^{2} + p^{2} T^{4} )^{4} \)
97 \( 1 - 360 T^{2} + 65929 T^{4} - 8182440 T^{6} + 834546960 T^{8} - 8182440 p^{2} T^{10} + 65929 p^{4} T^{12} - 360 p^{6} T^{14} + p^{8} T^{16} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.83529214180051570756799681045, −4.81081721033639541095872205646, −4.72545298102791268691171201093, −4.27263304044983162287177812681, −4.25749151379796398130779207759, −4.19265621924342431242707511972, −4.04807626796350292136293852205, −4.00982937557795730917977902309, −3.93826200667636714244217401464, −3.54728895740108376623479091716, −3.39523423587244600780480410898, −3.11740051329810810002918187879, −3.02656837639970305615287554712, −2.98827104967234392344874016957, −2.86562912005665874930117973554, −2.86457364642561709275964744117, −2.49843011870983795322708962396, −1.82068379500162274055569376777, −1.75092288025863658103510584705, −1.71782098077024742599213648851, −1.56450438086251025176548040031, −1.21532394690250189770591182579, −1.11328187380056148250192053040, −0.956360266069700077826968470920, −0.70357891176458529021146494111, 0.70357891176458529021146494111, 0.956360266069700077826968470920, 1.11328187380056148250192053040, 1.21532394690250189770591182579, 1.56450438086251025176548040031, 1.71782098077024742599213648851, 1.75092288025863658103510584705, 1.82068379500162274055569376777, 2.49843011870983795322708962396, 2.86457364642561709275964744117, 2.86562912005665874930117973554, 2.98827104967234392344874016957, 3.02656837639970305615287554712, 3.11740051329810810002918187879, 3.39523423587244600780480410898, 3.54728895740108376623479091716, 3.93826200667636714244217401464, 4.00982937557795730917977902309, 4.04807626796350292136293852205, 4.19265621924342431242707511972, 4.25749151379796398130779207759, 4.27263304044983162287177812681, 4.72545298102791268691171201093, 4.81081721033639541095872205646, 4.83529214180051570756799681045

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

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