Properties

Label 32-1400e16-1.1-c0e16-0-0
Degree $32$
Conductor $2.178\times 10^{50}$
Sign $1$
Analytic cond. $0.00322524$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 16-s − 8·49-s − 3·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s − 16·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 2·4-s + 16-s − 8·49-s − 3·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s − 16·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{48} \cdot 5^{32} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(0.00322524\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{48} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9003459537\)
\(L(\frac12)\) \(\approx\) \(0.9003459537\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
7 \( ( 1 + T^{2} )^{8} \)
good3 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
13 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
19 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
43 \( ( 1 + T^{2} )^{16} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
59 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
61 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
79 \( ( 1 + T^{2} )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
83 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.67121815734096461547153839704, −2.63828586661358771614611818293, −2.62240078266992149702370769786, −2.58875751020626439030698305832, −2.53143505801144383627364182538, −2.30080141689865138661017673037, −2.17091458178505453995293788491, −2.13390926212324365417884007073, −2.08441212485587964019482867358, −1.99529831607470987842968213744, −1.89028893538651765078316343077, −1.82697230194712186374138068627, −1.79634821642104132526691535871, −1.72390749674610302089145511011, −1.68188351044133965966852190134, −1.58018785390690581375111556196, −1.45499583956826926060293213301, −1.45264672726051789130269908628, −1.25997963002199621386917701032, −1.23065065796331416808026768510, −1.13247035947316375386959047337, −0.928049194437025365814701901397, −0.848698346682820657225015781619, −0.60415230265030939009640867070, −0.36957256646041690495853063504, 0.36957256646041690495853063504, 0.60415230265030939009640867070, 0.848698346682820657225015781619, 0.928049194437025365814701901397, 1.13247035947316375386959047337, 1.23065065796331416808026768510, 1.25997963002199621386917701032, 1.45264672726051789130269908628, 1.45499583956826926060293213301, 1.58018785390690581375111556196, 1.68188351044133965966852190134, 1.72390749674610302089145511011, 1.79634821642104132526691535871, 1.82697230194712186374138068627, 1.89028893538651765078316343077, 1.99529831607470987842968213744, 2.08441212485587964019482867358, 2.13390926212324365417884007073, 2.17091458178505453995293788491, 2.30080141689865138661017673037, 2.53143505801144383627364182538, 2.58875751020626439030698305832, 2.62240078266992149702370769786, 2.63828586661358771614611818293, 2.67121815734096461547153839704

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

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