Properties

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

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·7-s − 68·16-s + 112·25-s − 384·37-s + 64·43-s + 512·49-s + 320·67-s + 224·81-s − 2.17e3·112-s + 1.13e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 3.58e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 32/7·7-s − 4.25·16-s + 4.47·25-s − 10.3·37-s + 1.48·43-s + 10.4·49-s + 4.77·67-s + 2.76·81-s − 19.4·112-s + 9.38·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 20.4·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(3^{16} \cdot 5^{16} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(6.29083\)
\(L(\frac12)\)  \(\approx\)  \(6.29083\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad3 \( 1 - 224 T^{4} + 290 p^{4} T^{8} - 224 p^{8} T^{12} + p^{16} T^{16} \)
5 \( ( 1 - 56 T^{2} + 16 p^{3} T^{4} - 56 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
7 \( ( 1 - 16 T + 128 T^{2} - 208 T^{3} - 958 T^{4} - 208 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
good2 \( ( 1 + 17 T^{4} + p^{8} T^{8} )^{4} \)
11 \( ( 1 - 284 T^{2} + 40742 T^{4} - 284 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
13 \( ( 1 - 74400 T^{4} + 2876017858 T^{8} - 74400 p^{8} T^{12} + p^{16} T^{16} )^{2} \)
17 \( ( 1 + 280836 T^{4} + 33000880390 T^{8} + 280836 p^{8} T^{12} + p^{16} T^{16} )^{2} \)
19 \( ( 1 + 696 T^{2} + 371920 T^{4} + 696 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
23 \( ( 1 + 107644 T^{4} - 83134815354 T^{8} + 107644 p^{8} T^{12} + p^{16} T^{16} )^{2} \)
29 \( ( 1 + 2444 T^{2} + 2801222 T^{4} + 2444 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
31 \( ( 1 - 2596 T^{2} + 3425222 T^{4} - 2596 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
37 \( ( 1 + 96 T + 4608 T^{2} + 183264 T^{3} + 6996962 T^{4} + 183264 p^{2} T^{5} + 4608 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
41 \( ( 1 + 5716 T^{2} + 13775622 T^{4} + 5716 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
43 \( ( 1 - 16 T + 128 T^{2} - 2896 T^{3} - 2716702 T^{4} - 2896 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
47 \( ( 1 - 5511420 T^{4} + 7619082080518 T^{8} - 5511420 p^{8} T^{12} + p^{16} T^{16} )^{2} \)
53 \( ( 1 - 19870076 T^{4} + 198820934324166 T^{8} - 19870076 p^{8} T^{12} + p^{16} T^{16} )^{2} \)
59 \( ( 1 - 10776 T^{2} + 52783792 T^{4} - 10776 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
61 \( ( 1 - 8200 T^{2} + 36103376 T^{4} - 8200 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
67 \( ( 1 - 80 T + 3200 T^{2} + 17520 T^{3} - 22069342 T^{4} + 17520 p^{2} T^{5} + 3200 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
71 \( ( 1 - 17504 T^{2} + 126269762 T^{4} - 17504 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
73 \( ( 1 - 10432892 T^{4} + 418480444620678 T^{8} - 10432892 p^{8} T^{12} + p^{16} T^{16} )^{2} \)
79 \( ( 1 - 6736 T^{2} + p^{4} T^{4} )^{8} \)
83 \( ( 1 - 55419104 T^{4} + 5263624308845250 T^{8} - 55419104 p^{8} T^{12} + p^{16} T^{16} )^{2} \)
89 \( ( 1 - 1108 T^{2} - 101931898 T^{4} - 1108 p^{4} T^{6} + p^{8} T^{8} )^{4} \)
97 \( ( 1 + 284585220 T^{4} + 35721324461420038 T^{8} + 284585220 p^{8} T^{12} + p^{16} T^{16} )^{2} \)
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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

−4.08239974258686348291179754543, −3.80567008954487713144581923863, −3.66503462519461411284981128671, −3.53438050103299432174730216890, −3.53322753007468996011627988743, −3.35124504733449531519435395325, −3.29559659433136519559037254657, −3.25435627113390424588350071741, −3.22023649089247347666792604690, −2.67731720795549335619910659243, −2.66872122165220419197177896454, −2.58376431035778760152476442709, −2.43491242052798933074338252522, −2.34730008857012980853675029065, −2.21450153540500752366099952307, −1.98902820466104370209013269539, −1.90301172049601354602340813326, −1.86254738618732232498433600067, −1.69677607090898950991512393254, −1.52106135502945863759403470020, −1.43908391472416274587895741440, −1.17106605569589094207425958155, −0.72357733200806825756980922867, −0.69441233491994569917528715843, −0.24373590310307137208782387830, 0.24373590310307137208782387830, 0.69441233491994569917528715843, 0.72357733200806825756980922867, 1.17106605569589094207425958155, 1.43908391472416274587895741440, 1.52106135502945863759403470020, 1.69677607090898950991512393254, 1.86254738618732232498433600067, 1.90301172049601354602340813326, 1.98902820466104370209013269539, 2.21450153540500752366099952307, 2.34730008857012980853675029065, 2.43491242052798933074338252522, 2.58376431035778760152476442709, 2.66872122165220419197177896454, 2.67731720795549335619910659243, 3.22023649089247347666792604690, 3.25435627113390424588350071741, 3.29559659433136519559037254657, 3.35124504733449531519435395325, 3.53322753007468996011627988743, 3.53438050103299432174730216890, 3.66503462519461411284981128671, 3.80567008954487713144581923863, 4.08239974258686348291179754543

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

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