Properties

Degree 16
Conductor $ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} $
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  + 4·13-s − 2·16-s − 8·23-s − 24·29-s − 8·31-s + 4·37-s + 12·43-s + 12·47-s − 32·53-s − 16·59-s − 20·67-s − 36·73-s − 56·83-s − 72·89-s + 4·97-s + 24·103-s − 16·107-s − 32·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1.10·13-s − 1/2·16-s − 1.66·23-s − 4.45·29-s − 1.43·31-s + 0.657·37-s + 1.82·43-s + 1.75·47-s − 4.39·53-s − 2.08·59-s − 2.44·67-s − 4.21·73-s − 6.14·83-s − 7.63·89-s + 0.406·97-s + 2.36·103-s − 1.54·107-s − 3.01·113-s + 1.09·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $2.434823699$
$L(\frac12)$  $\approx$  $2.434823699$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( 1 - 12 T^{2} + 40 T^{4} - 260 T^{6} + 15086 T^{8} - 260 p^{2} T^{10} + 40 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 4 T + 8 T^{2} - 28 T^{3} + 80 T^{4} - 372 T^{5} + 1240 T^{6} - 6412 T^{7} + 34174 T^{8} - 6412 p T^{9} + 1240 p^{2} T^{10} - 372 p^{3} T^{11} + 80 p^{4} T^{12} - 28 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 - 64 T^{3} - 252 T^{4} + 960 T^{5} + 2048 T^{6} + 6016 T^{7} - 17786 T^{8} + 6016 p T^{9} + 2048 p^{2} T^{10} + 960 p^{3} T^{11} - 252 p^{4} T^{12} - 64 p^{5} T^{13} + p^{8} T^{16} \)
19 \( ( 1 - p T^{2} )^{8} \)
23 \( 1 + 8 T + 32 T^{2} - 8 T^{3} - 924 T^{4} - 5032 T^{5} - 10656 T^{6} + 28456 T^{7} + 420614 T^{8} + 28456 p T^{9} - 10656 p^{2} T^{10} - 5032 p^{3} T^{11} - 924 p^{4} T^{12} - 8 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 + 12 T + 78 T^{2} + 252 T^{3} + 738 T^{4} + 252 p T^{5} + 78 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 4 T + 110 T^{2} + 356 T^{3} + 4930 T^{4} + 356 p T^{5} + 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 4 T + 8 T^{2} - 220 T^{3} - 48 T^{4} + 8684 T^{5} - 10152 T^{6} + 208404 T^{7} - 3196994 T^{8} + 208404 p T^{9} - 10152 p^{2} T^{10} + 8684 p^{3} T^{11} - 48 p^{4} T^{12} - 220 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 76 T^{2} + 5480 T^{4} - 315940 T^{6} + 12648846 T^{8} - 315940 p^{2} T^{10} + 5480 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 12 T + 72 T^{2} - 300 T^{3} - 1008 T^{4} + 17988 T^{5} - 98280 T^{6} + 509316 T^{7} - 2377282 T^{8} + 509316 p T^{9} - 98280 p^{2} T^{10} + 17988 p^{3} T^{11} - 1008 p^{4} T^{12} - 300 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 12 T + 72 T^{2} - 220 T^{3} - 2352 T^{4} + 10116 T^{5} + 72152 T^{6} - 1599596 T^{7} + 18271774 T^{8} - 1599596 p T^{9} + 72152 p^{2} T^{10} + 10116 p^{3} T^{11} - 2352 p^{4} T^{12} - 220 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 32 T + 512 T^{2} + 5792 T^{3} + 60388 T^{4} + 619872 T^{5} + 5690880 T^{6} + 44817632 T^{7} + 328258854 T^{8} + 44817632 p T^{9} + 5690880 p^{2} T^{10} + 619872 p^{3} T^{11} + 60388 p^{4} T^{12} + 5792 p^{5} T^{13} + 512 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \)
59 \( ( 1 + 8 T + 148 T^{2} + 520 T^{3} + 8710 T^{4} + 520 p T^{5} + 148 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 226 T^{2} + 16 T^{3} + 20162 T^{4} + 16 p T^{5} + 226 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( 1 + 20 T + 200 T^{2} + 1332 T^{3} + 9104 T^{4} + 102596 T^{5} + 1118232 T^{6} + 8938244 T^{7} + 69729150 T^{8} + 8938244 p T^{9} + 1118232 p^{2} T^{10} + 102596 p^{3} T^{11} + 9104 p^{4} T^{12} + 1332 p^{5} T^{13} + 200 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 216 T^{2} + 31388 T^{4} - 3350120 T^{6} + 263319558 T^{8} - 3350120 p^{2} T^{10} + 31388 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 + 36 T + 648 T^{2} + 8620 T^{3} + 98192 T^{4} + 980116 T^{5} + 8807960 T^{6} + 73999132 T^{7} + 618591198 T^{8} + 73999132 p T^{9} + 8807960 p^{2} T^{10} + 980116 p^{3} T^{11} + 98192 p^{4} T^{12} + 8620 p^{5} T^{13} + 648 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 - 280 T^{2} + 32732 T^{4} - 1914792 T^{6} + 96372294 T^{8} - 1914792 p^{2} T^{10} + 32732 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 + 56 T + 1568 T^{2} + 30056 T^{3} + 449924 T^{4} + 5649064 T^{5} + 62548320 T^{6} + 632947128 T^{7} + 5962291750 T^{8} + 632947128 p T^{9} + 62548320 p^{2} T^{10} + 5649064 p^{3} T^{11} + 449924 p^{4} T^{12} + 30056 p^{5} T^{13} + 1568 p^{6} T^{14} + 56 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 + 36 T + 734 T^{2} + 10004 T^{3} + 106562 T^{4} + 10004 p T^{5} + 734 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 4 T + 8 T^{2} - 1852 T^{3} + 7856 T^{4} + 48156 T^{5} + 1459480 T^{6} - 10407580 T^{7} - 90680354 T^{8} - 10407580 p T^{9} + 1459480 p^{2} T^{10} + 48156 p^{3} T^{11} + 7856 p^{4} T^{12} - 1852 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
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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

−3.60083582505337136073993184971, −3.58240181729126700598357497036, −3.44318323232005031188363056223, −3.16706535966035647575700633929, −3.03654162561147083392895078203, −2.96337884542381945494313459814, −2.88299047034554229672844615564, −2.83212625491807509768174662860, −2.62708604389060384656534928051, −2.60741457284700853663918369216, −2.57550672349662322021312789467, −2.26973901088499906145189652975, −1.97322765638374896370438579363, −1.81399755582543843464312765105, −1.71800518527658592948773374683, −1.69074683003654057948147483787, −1.54561443447422052246043433560, −1.53241031809631857285470866917, −1.50078223723858994580341304764, −1.31341415813654528388191711476, −1.07821240856353839095209798899, −0.49979045000293173616935033280, −0.38493884641783686940225514723, −0.30817707298343377363356954312, −0.25779115565137003681502248246, 0.25779115565137003681502248246, 0.30817707298343377363356954312, 0.38493884641783686940225514723, 0.49979045000293173616935033280, 1.07821240856353839095209798899, 1.31341415813654528388191711476, 1.50078223723858994580341304764, 1.53241031809631857285470866917, 1.54561443447422052246043433560, 1.69074683003654057948147483787, 1.71800518527658592948773374683, 1.81399755582543843464312765105, 1.97322765638374896370438579363, 2.26973901088499906145189652975, 2.57550672349662322021312789467, 2.60741457284700853663918369216, 2.62708604389060384656534928051, 2.83212625491807509768174662860, 2.88299047034554229672844615564, 2.96337884542381945494313459814, 3.03654162561147083392895078203, 3.16706535966035647575700633929, 3.44318323232005031188363056223, 3.58240181729126700598357497036, 3.60083582505337136073993184971

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

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