Properties

Degree 32
Conductor $ 2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·7-s + 16·11-s + 40·23-s + 32·37-s + 16·43-s + 32·49-s + 24·53-s + 32·67-s − 64·71-s + 128·77-s − 4·81-s + 8·107-s + 8·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 320·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 3.02·7-s + 4.82·11-s + 8.34·23-s + 5.26·37-s + 2.43·43-s + 32/7·49-s + 3.29·53-s + 3.90·67-s − 7.59·71-s + 14.5·77-s − 4/9·81-s + 0.773·107-s + 0.752·113-s + 2.90·121-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 + 25.2·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1680} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{64} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(229.3628801\)
\(L(\frac12)\)  \(\approx\)  \(229.3628801\)
\(L(\frac{3}{2})\)   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 32. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{4} )^{4} \)
5 \( 1 + 28 T^{4} - 256 T^{6} - 26 T^{8} - 256 p^{2} T^{10} + 28 p^{4} T^{12} + p^{8} T^{16} \)
7 \( 1 - 8 T + 32 T^{2} - 88 T^{3} + 4 p^{2} T^{4} - 248 T^{5} - 416 T^{6} + 2840 T^{7} - 8634 T^{8} + 2840 p T^{9} - 416 p^{2} T^{10} - 248 p^{3} T^{11} + 4 p^{6} T^{12} - 88 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
good11 \( ( 1 - 4 T + 32 T^{2} - 68 T^{3} + 402 T^{4} - 68 p T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
13 \( 1 + 424 T^{4} + 47004 T^{8} - 12160488 T^{12} - 4129271418 T^{16} - 12160488 p^{4} T^{20} + 47004 p^{8} T^{24} + 424 p^{12} T^{28} + p^{16} T^{32} \)
17 \( 1 + 120 T^{4} + 166556 T^{8} - 9625784 T^{12} + 12237871174 T^{16} - 9625784 p^{4} T^{20} + 166556 p^{8} T^{24} + 120 p^{12} T^{28} + p^{16} T^{32} \)
19 \( ( 1 + 48 T^{2} + 1524 T^{4} + 40016 T^{6} + 818246 T^{8} + 40016 p^{2} T^{10} + 1524 p^{4} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 20 T + 200 T^{2} - 1516 T^{3} + 10388 T^{4} - 65340 T^{5} + 378328 T^{6} - 2076676 T^{7} + 10539814 T^{8} - 2076676 p T^{9} + 378328 p^{2} T^{10} - 65340 p^{3} T^{11} + 10388 p^{4} T^{12} - 1516 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 184 T^{2} + 15868 T^{4} - 835400 T^{6} + 29324070 T^{8} - 835400 p^{2} T^{10} + 15868 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 128 T^{2} + 9396 T^{4} - 463040 T^{6} + 16703398 T^{8} - 463040 p^{2} T^{10} + 9396 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 16 T + 128 T^{2} - 944 T^{3} + 8860 T^{4} - 73552 T^{5} + 488320 T^{6} - 3175280 T^{7} + 20212134 T^{8} - 3175280 p T^{9} + 488320 p^{2} T^{10} - 73552 p^{3} T^{11} + 8860 p^{4} T^{12} - 944 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 144 T^{2} + 316 p T^{4} - 823792 T^{6} + 38320198 T^{8} - 823792 p^{2} T^{10} + 316 p^{5} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 8 T + 32 T^{2} - 280 T^{3} + 2788 T^{4} - 14232 T^{5} + 63840 T^{6} - 569416 T^{7} + 5017638 T^{8} - 569416 p T^{9} + 63840 p^{2} T^{10} - 14232 p^{3} T^{11} + 2788 p^{4} T^{12} - 280 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( 1 + 3784 T^{4} + 1124764 T^{8} + 9138019192 T^{12} + 63538455194182 T^{16} + 9138019192 p^{4} T^{20} + 1124764 p^{8} T^{24} + 3784 p^{12} T^{28} + p^{16} T^{32} \)
53 \( ( 1 - 12 T + 72 T^{2} - 572 T^{3} + 1780 T^{4} + 1436 T^{5} + 18200 T^{6} - 441076 T^{7} + 6254598 T^{8} - 441076 p T^{9} + 18200 p^{2} T^{10} + 1436 p^{3} T^{11} + 1780 p^{4} T^{12} - 572 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 312 T^{2} + 49660 T^{4} + 5040712 T^{6} + 354176614 T^{8} + 5040712 p^{2} T^{10} + 49660 p^{4} T^{12} + 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 200 T^{2} + 17532 T^{4} - 857912 T^{6} + 37932838 T^{8} - 857912 p^{2} T^{10} + 17532 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 16 T + 128 T^{2} - 1424 T^{3} + 22436 T^{4} - 211280 T^{5} + 1522560 T^{6} - 15870032 T^{7} + 163564774 T^{8} - 15870032 p T^{9} + 1522560 p^{2} T^{10} - 211280 p^{3} T^{11} + 22436 p^{4} T^{12} - 1424 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 16 T + 232 T^{2} + 2376 T^{3} + 21730 T^{4} + 2376 p T^{5} + 232 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
73 \( 1 - 15256 T^{4} + 80862300 T^{8} + 94053698264 T^{12} - 2362018367550906 T^{16} + 94053698264 p^{4} T^{20} + 80862300 p^{8} T^{24} - 15256 p^{12} T^{28} + p^{16} T^{32} \)
79 \( ( 1 - 312 T^{2} + 58396 T^{4} - 7293320 T^{6} + 672141766 T^{8} - 7293320 p^{2} T^{10} + 58396 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( 1 + 5000 T^{4} + 95818588 T^{8} + 596752860728 T^{12} + 4571727903671302 T^{16} + 596752860728 p^{4} T^{20} + 95818588 p^{8} T^{24} + 5000 p^{12} T^{28} + p^{16} T^{32} \)
89 \( ( 1 + 576 T^{2} + 155068 T^{4} + 25338304 T^{6} + 2740378246 T^{8} + 25338304 p^{2} T^{10} + 155068 p^{4} T^{12} + 576 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( 1 - 55064 T^{4} + 1465436892 T^{8} - 24336256217256 T^{12} + 274732504520067270 T^{16} - 24336256217256 p^{4} T^{20} + 1465436892 p^{8} T^{24} - 55064 p^{12} T^{28} + p^{16} T^{32} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.31346891834334979940297797602, −2.16869179092582434265487492672, −2.11787839723481740388908280199, −2.10667435463428702124867969665, −2.08790837586906422541795255303, −2.02541680437625225546900250374, −1.96434009582384106667069021105, −1.96199282817752544882982398097, −1.68426389345896051208933749877, −1.46351429508349018506931017905, −1.31898632942659540312711397691, −1.30892733756317159385650168993, −1.29168434142689295102366881641, −1.29101926451057473091092356813, −1.26477119409214139475181535331, −1.17258642491509972691804441691, −1.07539600389118560525244872380, −1.06581015204560507918505721862, −1.02205076281411068463562019929, −0.884317354470044676764357256191, −0.834172367789482434231269979883, −0.60656275943648619892643756376, −0.56144630878097495566190186065, −0.40897104025860906953761123947, −0.16759613966904498662745504018, 0.16759613966904498662745504018, 0.40897104025860906953761123947, 0.56144630878097495566190186065, 0.60656275943648619892643756376, 0.834172367789482434231269979883, 0.884317354470044676764357256191, 1.02205076281411068463562019929, 1.06581015204560507918505721862, 1.07539600389118560525244872380, 1.17258642491509972691804441691, 1.26477119409214139475181535331, 1.29101926451057473091092356813, 1.29168434142689295102366881641, 1.30892733756317159385650168993, 1.31898632942659540312711397691, 1.46351429508349018506931017905, 1.68426389345896051208933749877, 1.96199282817752544882982398097, 1.96434009582384106667069021105, 2.02541680437625225546900250374, 2.08790837586906422541795255303, 2.10667435463428702124867969665, 2.11787839723481740388908280199, 2.16869179092582434265487492672, 2.31346891834334979940297797602

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

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