Properties

Label 32-300e16-1.1-c1e16-0-0
Degree $32$
Conductor $4.305\times 10^{39}$
Sign $1$
Analytic cond. $1.17590\times 10^{6}$
Root an. cond. $1.54774$
Motivic weight $1$
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 − 2·9-s + 7·16-s − 4·36-s − 36·49-s − 40·61-s + 24·64-s − 13·81-s − 88·109-s − 60·121-s + 127-s + 131-s + 137-s + 139-s − 14·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 124·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 72·196-s + ⋯
L(s)  = 1  + 4-s − 2/3·9-s + 7/4·16-s − 2/3·36-s − 5.14·49-s − 5.12·61-s + 3·64-s − 1.44·81-s − 8.42·109-s − 5.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 5.14·196-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(1.17590\times 10^{6}\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.09453237333\)
\(L(\frac12)\) \(\approx\) \(0.09453237333\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} - p T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \)
3 \( ( 1 + T^{2} + 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \)
5 \( 1 \)
good7 \( ( 1 + 9 T^{2} + 108 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
11 \( ( 1 + 15 T^{2} + 288 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
13 \( ( 1 - 31 T^{2} + 568 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
17 \( ( 1 + 9 T^{2} - 232 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
19 \( ( 1 - 50 T^{2} + 1183 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
23 \( ( 1 - 64 T^{2} + 1918 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
29 \( ( 1 - 80 T^{2} + 3118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
31 \( ( 1 - 69 T^{2} + 2856 T^{4} - 69 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
37 \( ( 1 - 64 T^{2} + 3598 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
41 \( ( 1 - 49 T^{2} + 1656 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
43 \( ( 1 + 101 T^{2} + 5008 T^{4} + 101 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
47 \( ( 1 - 136 T^{2} + 8878 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
53 \( ( 1 + 176 T^{2} + 13198 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
59 \( ( 1 + p T^{2} )^{16} \)
61 \( ( 1 + 5 T + 118 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{8} \)
67 \( ( 1 + 234 T^{2} + 22503 T^{4} + 234 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
71 \( ( 1 + 9918 T^{4} + p^{4} T^{8} )^{4} \)
73 \( ( 1 - 231 T^{2} + 23168 T^{4} - 231 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
79 \( ( 1 - 240 T^{2} + 26718 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
83 \( ( 1 - 319 T^{2} + 39208 T^{4} - 319 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
89 \( ( 1 - 305 T^{2} + 39088 T^{4} - 305 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
97 \( ( 1 - 199 T^{2} + 27888 T^{4} - 199 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
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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

−3.25421294065577847246175339931, −3.14565147961221435013572818879, −3.11516937366209048125128476560, −3.06218842420574496143559062842, −3.04058434144265175249116528068, −2.89066591929788769325138838931, −2.83637714172766041345201897268, −2.72226255833081981477679406312, −2.49672116770888570075259609020, −2.46468720516713949845520552896, −2.44102628477068629742108176702, −2.36847889215001632135701481104, −2.31353251519839730619185020203, −1.98774797936977134987421084730, −1.81490191998263523169668748656, −1.75914422939315494596899954358, −1.64303959709078520672530672139, −1.55348969953339636927546846569, −1.43465821557815491582527101258, −1.37441001210487316748654423987, −1.28747247565519374857029627366, −1.16975258407951238491011294602, −0.805415195513620145705878779717, −0.42493530527779318952078622417, −0.04703137516130554152457637370, 0.04703137516130554152457637370, 0.42493530527779318952078622417, 0.805415195513620145705878779717, 1.16975258407951238491011294602, 1.28747247565519374857029627366, 1.37441001210487316748654423987, 1.43465821557815491582527101258, 1.55348969953339636927546846569, 1.64303959709078520672530672139, 1.75914422939315494596899954358, 1.81490191998263523169668748656, 1.98774797936977134987421084730, 2.31353251519839730619185020203, 2.36847889215001632135701481104, 2.44102628477068629742108176702, 2.46468720516713949845520552896, 2.49672116770888570075259609020, 2.72226255833081981477679406312, 2.83637714172766041345201897268, 2.89066591929788769325138838931, 3.04058434144265175249116528068, 3.06218842420574496143559062842, 3.11516937366209048125128476560, 3.14565147961221435013572818879, 3.25421294065577847246175339931

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

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