Properties

Label 32-33e32-1.1-c0e16-0-0
Degree $32$
Conductor $3.912\times 10^{48}$
Sign $1$
Analytic cond. $5.79375\times 10^{-5}$
Root an. cond. $0.737212$
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  − 2·3-s − 2·5-s + 3·9-s + 4·15-s − 16-s + 25-s − 2·27-s + 2·31-s + 4·37-s − 6·45-s − 2·47-s + 2·48-s − 4·53-s + 2·59-s + 8·67-s + 4·71-s − 2·75-s + 2·80-s + 81-s − 4·93-s + 2·97-s − 2·103-s − 8·111-s − 2·113-s − 2·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 2·3-s − 2·5-s + 3·9-s + 4·15-s − 16-s + 25-s − 2·27-s + 2·31-s + 4·37-s − 6·45-s − 2·47-s + 2·48-s − 4·53-s + 2·59-s + 8·67-s + 4·71-s − 2·75-s + 2·80-s + 81-s − 4·93-s + 2·97-s − 2·103-s − 8·111-s − 2·113-s − 2·125-s + 127-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 11^{32}\)
Sign: $1$
Analytic conductor: \(5.79375\times 10^{-5}\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 11^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1775874017\)
\(L(\frac12)\) \(\approx\) \(0.1775874017\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
11 \( 1 \)
good2 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
7 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
23 \( ( 1 - T^{2} + T^{4} )^{8} \)
29 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{4} \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
43 \( ( 1 - T + T^{2} )^{8}( 1 + T + T^{2} )^{8} \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
53 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{4} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
61 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
67 \( ( 1 - T )^{16}( 1 + T + T^{2} )^{8} \)
71 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{4} \)
73 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
79 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
83 \( 1 + T^{4} - T^{12} - T^{16} - T^{20} + T^{28} + T^{32} \)
89 \( ( 1 + T^{2} )^{16} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
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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

−2.81010196425896005412242348722, −2.75035842885963088004696546835, −2.70220341436518213873675353123, −2.53975633314205746964399173467, −2.50810393779430471344256421674, −2.45605676292246374281323341074, −2.33830857970292632373847837164, −2.28767458697672807493993280846, −2.17689025838903540195740698657, −2.10278417985452916650104872269, −2.09084264478343865385506983788, −2.04083923661676067610064735082, −1.99667463695322544675256769664, −1.80493254958784447169420725039, −1.65494580223565879473431541364, −1.48115754420001672506496018547, −1.39226227724889897118245719087, −1.38697578060642748414256129423, −1.07513108890654075612351258585, −1.06388463133852421258193741947, −1.03721026632452037312339256611, −0.980082450101078726749206518329, −0.909089372346202365226210124633, −0.71242096520896867364893159056, −0.47408370384288265223877146384, 0.47408370384288265223877146384, 0.71242096520896867364893159056, 0.909089372346202365226210124633, 0.980082450101078726749206518329, 1.03721026632452037312339256611, 1.06388463133852421258193741947, 1.07513108890654075612351258585, 1.38697578060642748414256129423, 1.39226227724889897118245719087, 1.48115754420001672506496018547, 1.65494580223565879473431541364, 1.80493254958784447169420725039, 1.99667463695322544675256769664, 2.04083923661676067610064735082, 2.09084264478343865385506983788, 2.10278417985452916650104872269, 2.17689025838903540195740698657, 2.28767458697672807493993280846, 2.33830857970292632373847837164, 2.45605676292246374281323341074, 2.50810393779430471344256421674, 2.53975633314205746964399173467, 2.70220341436518213873675353123, 2.75035842885963088004696546835, 2.81010196425896005412242348722

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

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