Properties

Degree 32
Conductor $ 2^{96} \cdot 5^{16} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 992·9-s + 440·25-s − 1.00e5·41-s + 1.98e5·49-s + 3.37e5·81-s − 4.59e5·89-s + 2.30e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.04e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 4.36e5·225-s + ⋯
L(s)  = 1  − 4.08·9-s + 0.140·25-s − 9.32·41-s + 11.8·49-s + 5.71·81-s − 6.14·89-s + 14.3·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 10.9·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s − 0.574·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{96} \cdot 5^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((32,\ 2^{96} \cdot 5^{16} ,\ ( \ : [5/2]^{16} ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(2.842403078\)
\(L(\frac12)\)  \(\approx\)  \(2.842403078\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - 44 p T^{2} - 48914 p^{3} T^{4} - 44 p^{11} T^{6} + p^{20} T^{8} )^{2} \)
good3 \( ( 1 + 248 T^{2} + 23110 p T^{4} + 248 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
7 \( ( 1 - 49648 T^{2} + 1179577874 T^{4} - 49648 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
11 \( ( 1 - 577284 T^{2} + 134071408310 T^{4} - 577284 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
13 \( ( 1 + 1011892 T^{2} + 531287817014 T^{4} + 1011892 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
17 \( ( 1 + 554828 T^{2} + 3105096833510 T^{4} + 554828 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
19 \( ( 1 - 8279076 T^{2} + 29313371734870 T^{4} - 8279076 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
23 \( ( 1 - 265392 T^{2} + 72532989580114 T^{4} - 265392 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
29 \( ( 1 - 18262996 T^{2} + 467877918484406 T^{4} - 18262996 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
31 \( ( 1 + 39407004 T^{2} + 689994929185606 T^{4} + 39407004 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
37 \( ( 1 + 171738148 T^{2} + 16366808202619574 T^{4} + 171738148 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
41 \( ( 1 + 12544 T + 129356290 T^{2} + 12544 p^{5} T^{3} + p^{10} T^{4} )^{8} \)
43 \( ( 1 + 515310072 T^{2} + 108506205300479794 T^{4} + 515310072 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
47 \( ( 1 - 112031408 T^{2} + 100831510268064114 T^{4} - 112031408 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
53 \( ( 1 + 823212052 T^{2} + 458582260389137174 T^{4} + 823212052 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
59 \( ( 1 - 2409567364 T^{2} + 2462718757251169526 T^{4} - 2409567364 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
61 \( ( 1 - 2000262204 T^{2} + 2103804668157499606 T^{4} - 2000262204 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
67 \( ( 1 + 3461519512 T^{2} + 6000732802449327570 T^{4} + 3461519512 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
71 \( ( 1 + 823411004 T^{2} + 3069578968927394406 T^{4} + 823411004 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
73 \( ( 1 - 4148991508 T^{2} + 8650887997979942790 T^{4} - 4148991508 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
79 \( ( 1 + 2847963996 T^{2} + 15373907451301926406 T^{4} + 2847963996 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
83 \( ( 1 + 7032005368 T^{2} + 39846447958779448850 T^{4} + 7032005368 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
89 \( ( 1 + 57412 T + 10662063350 T^{2} + 57412 p^{5} T^{3} + p^{10} T^{4} )^{8} \)
97 \( ( 1 - 31837059828 T^{2} + \)\(40\!\cdots\!94\)\( T^{4} - 31837059828 p^{10} T^{6} + p^{20} T^{8} )^{4} \)
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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

−2.27463632637163423282804799458, −2.19538172380811106092459178259, −2.16858178137781383336723912090, −2.14639742736100374746648384997, −2.11130631221896629197328060309, −1.86374653166404933977668404552, −1.86325758900006500940121914949, −1.66953869193184611159839925655, −1.61890108076752489292556942430, −1.47091482650619724902993193984, −1.43517707371419276178276038221, −1.32179712212076168431779208156, −1.17691565364933442625582831168, −1.16876675346556014405129366227, −0.952449072385739289890792611965, −0.923662891989800893663854244573, −0.800509889831658151805998370560, −0.72437881763799296003080238129, −0.51842897044945029400819132120, −0.48424041414106230515666908283, −0.42428013887484635134782145604, −0.29025436164926788000856105119, −0.26080891014079971409091142885, −0.23327935948299229808759598813, −0.06849415050395067155260929386, 0.06849415050395067155260929386, 0.23327935948299229808759598813, 0.26080891014079971409091142885, 0.29025436164926788000856105119, 0.42428013887484635134782145604, 0.48424041414106230515666908283, 0.51842897044945029400819132120, 0.72437881763799296003080238129, 0.800509889831658151805998370560, 0.923662891989800893663854244573, 0.952449072385739289890792611965, 1.16876675346556014405129366227, 1.17691565364933442625582831168, 1.32179712212076168431779208156, 1.43517707371419276178276038221, 1.47091482650619724902993193984, 1.61890108076752489292556942430, 1.66953869193184611159839925655, 1.86325758900006500940121914949, 1.86374653166404933977668404552, 2.11130631221896629197328060309, 2.14639742736100374746648384997, 2.16858178137781383336723912090, 2.19538172380811106092459178259, 2.27463632637163423282804799458

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

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