Properties

Label 32-300e16-1.1-c7e16-0-0
Degree $32$
Conductor $4.305\times 10^{39}$
Sign $1$
Analytic cond. $3.53983\times 10^{31}$
Root an. cond. $9.68067$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.72e5·31-s + 1.27e7·61-s + 2.67e6·81-s + 9.32e7·121-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  + 5.26·31-s + 7.19·61-s + 0.559·81-s + 4.78·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(3.53983\times 10^{31}\)
Root analytic conductor: \(9.68067\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [7/2]^{16} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(0.1892358896\)
\(L(\frac12)\) \(\approx\) \(0.1892358896\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 297116 p^{2} T^{4} + 1708846246 p^{8} T^{8} - 297116 p^{30} T^{12} + p^{56} T^{16} \)
5 \( 1 \)
good7 \( ( 1 + 738644885092 T^{4} + \)\(55\!\cdots\!18\)\( T^{8} + 738644885092 p^{28} T^{12} + p^{56} T^{16} )^{2} \)
11 \( ( 1 - 23310764 T^{2} + 422598397878006 T^{4} - 23310764 p^{14} T^{6} + p^{28} T^{8} )^{4} \)
13 \( ( 1 + 4556964764612452 T^{4} + \)\(27\!\cdots\!38\)\( p^{4} T^{8} + 4556964764612452 p^{28} T^{12} + p^{56} T^{16} )^{2} \)
17 \( ( 1 - 578564106573898364 T^{4} + \)\(14\!\cdots\!06\)\( T^{8} - 578564106573898364 p^{28} T^{12} + p^{56} T^{16} )^{2} \)
19 \( ( 1 - 1480918988 T^{2} + 1941397902416113878 T^{4} - 1480918988 p^{14} T^{6} + p^{28} T^{8} )^{4} \)
23 \( ( 1 + 25553954396507481316 T^{4} + \)\(41\!\cdots\!26\)\( T^{8} + 25553954396507481316 p^{28} T^{12} + p^{56} T^{16} )^{2} \)
29 \( ( 1 + 24240643316 T^{2} + \)\(68\!\cdots\!26\)\( T^{4} + 24240643316 p^{14} T^{6} + p^{28} T^{8} )^{4} \)
31 \( ( 1 - 109088 T + 27875922558 T^{2} - 109088 p^{7} T^{3} + p^{14} T^{4} )^{8} \)
37 \( ( 1 + \)\(13\!\cdots\!16\)\( T^{4} + \)\(18\!\cdots\!06\)\( T^{8} + \)\(13\!\cdots\!16\)\( p^{28} T^{12} + p^{56} T^{16} )^{2} \)
41 \( ( 1 + 201323855356 T^{2} + \)\(84\!\cdots\!06\)\( T^{4} + 201323855356 p^{14} T^{6} + p^{28} T^{8} )^{4} \)
43 \( ( 1 - \)\(10\!\cdots\!08\)\( T^{4} + \)\(92\!\cdots\!18\)\( T^{8} - \)\(10\!\cdots\!08\)\( p^{28} T^{12} + p^{56} T^{16} )^{2} \)
47 \( ( 1 + \)\(83\!\cdots\!56\)\( T^{4} + \)\(29\!\cdots\!06\)\( T^{8} + \)\(83\!\cdots\!56\)\( p^{28} T^{12} + p^{56} T^{16} )^{2} \)
53 \( ( 1 + \)\(36\!\cdots\!56\)\( T^{4} + \)\(63\!\cdots\!06\)\( T^{8} + \)\(36\!\cdots\!56\)\( p^{28} T^{12} + p^{56} T^{16} )^{2} \)
59 \( ( 1 + 7552112348396 T^{2} + \)\(25\!\cdots\!26\)\( T^{4} + 7552112348396 p^{14} T^{6} + p^{28} T^{8} )^{4} \)
61 \( ( 1 - 1594108 T + 5501637390558 T^{2} - 1594108 p^{7} T^{3} + p^{14} T^{4} )^{8} \)
67 \( ( 1 - \)\(28\!\cdots\!68\)\( T^{4} + \)\(29\!\cdots\!38\)\( T^{8} - \)\(28\!\cdots\!68\)\( p^{28} T^{12} + p^{56} T^{16} )^{2} \)
71 \( ( 1 - 30191916548444 T^{2} + \)\(38\!\cdots\!46\)\( T^{4} - 30191916548444 p^{14} T^{6} + p^{28} T^{8} )^{4} \)
73 \( ( 1 - \)\(42\!\cdots\!44\)\( T^{4} + \)\(73\!\cdots\!46\)\( T^{8} - \)\(42\!\cdots\!44\)\( p^{28} T^{12} + p^{56} T^{16} )^{2} \)
79 \( ( 1 - 67295855245244 T^{2} + \)\(18\!\cdots\!46\)\( T^{4} - 67295855245244 p^{14} T^{6} + p^{28} T^{8} )^{4} \)
83 \( ( 1 - \)\(76\!\cdots\!64\)\( T^{4} + \)\(62\!\cdots\!06\)\( T^{8} - \)\(76\!\cdots\!64\)\( p^{28} T^{12} + p^{56} T^{16} )^{2} \)
89 \( ( 1 + 136054353608996 T^{2} + \)\(84\!\cdots\!86\)\( T^{4} + 136054353608996 p^{14} T^{6} + p^{28} T^{8} )^{4} \)
97 \( ( 1 + \)\(17\!\cdots\!32\)\( T^{4} + \)\(13\!\cdots\!78\)\( T^{8} + \)\(17\!\cdots\!32\)\( p^{28} T^{12} + p^{56} T^{16} )^{2} \)
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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

−2.22478261057933853117071684318, −2.00865529148380358338303094750, −1.98256788533433913607024239800, −1.93246711820009649490783568032, −1.87075436122325522718108511206, −1.58940758707628016131669631518, −1.52299343315089617296054060062, −1.43630260126463667640317845530, −1.32222310641343908989677116673, −1.31122683927920804834651958202, −1.22184016080136453435783432396, −1.03983981346215821815666933255, −0.972905698602943728527516158442, −0.926755043266091466508941395935, −0.915939254120453771565684925970, −0.895852094849450602243651363148, −0.826455975817323715099989942604, −0.75215817795625197351574627996, −0.58960208945511585640683955098, −0.56910825273293113330278835741, −0.38681080182567198022772835507, −0.31572827895328921975065278808, −0.20326632402548862920240690671, −0.086364676113700296288276749004, −0.01547527579046980177481480587, 0.01547527579046980177481480587, 0.086364676113700296288276749004, 0.20326632402548862920240690671, 0.31572827895328921975065278808, 0.38681080182567198022772835507, 0.56910825273293113330278835741, 0.58960208945511585640683955098, 0.75215817795625197351574627996, 0.826455975817323715099989942604, 0.895852094849450602243651363148, 0.915939254120453771565684925970, 0.926755043266091466508941395935, 0.972905698602943728527516158442, 1.03983981346215821815666933255, 1.22184016080136453435783432396, 1.31122683927920804834651958202, 1.32222310641343908989677116673, 1.43630260126463667640317845530, 1.52299343315089617296054060062, 1.58940758707628016131669631518, 1.87075436122325522718108511206, 1.93246711820009649490783568032, 1.98256788533433913607024239800, 2.00865529148380358338303094750, 2.22478261057933853117071684318

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

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