Properties

Label 32-21e32-1.1-c3e16-0-1
Degree $32$
Conductor $2.047\times 10^{42}$
Sign $1$
Analytic cond. $4.41442\times 10^{22}$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 32·4-s + 478·16-s − 980·25-s + 2.39e3·37-s + 328·43-s + 5.78e3·64-s − 616·67-s + 4.35e3·79-s − 3.13e4·100-s + 2.15e3·109-s + 6.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 7.65e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.00e3·169-s + 1.04e4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 4·4-s + 7.46·16-s − 7.83·25-s + 10.6·37-s + 1.16·43-s + 11.2·64-s − 1.12·67-s + 6.19·79-s − 31.3·100-s + 1.89·109-s + 4.81·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 42.5·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4.09·169-s + 4.65·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(4.41442\times 10^{22}\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 7^{32} ,\ ( \ : [3/2]^{16} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(296.7520810\)
\(L(\frac12)\) \(\approx\) \(296.7520810\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - p^{4} T^{2} + 145 T^{4} - 829 p T^{6} + 4265 p^{2} T^{8} - 829 p^{7} T^{10} + 145 p^{12} T^{12} - p^{22} T^{14} + p^{24} T^{16} )^{2} \)
5 \( ( 1 + 98 p T^{2} + 5557 p^{2} T^{4} + 27302758 T^{6} + 3917065816 T^{8} + 27302758 p^{6} T^{10} + 5557 p^{14} T^{12} + 98 p^{19} T^{14} + p^{24} T^{16} )^{2} \)
11 \( ( 1 - 3202 T^{2} + 8455357 T^{4} - 16341433166 T^{6} + 23699042774456 T^{8} - 16341433166 p^{6} T^{10} + 8455357 p^{12} T^{12} - 3202 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
13 \( ( 1 - 4502 T^{2} + 10097461 T^{4} - 24107307194 T^{6} + 60089639013832 T^{8} - 24107307194 p^{6} T^{10} + 10097461 p^{12} T^{12} - 4502 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
17 \( ( 1 + 16948 T^{2} + 203031496 T^{4} + 1520947046428 T^{6} + 8936065158707086 T^{8} + 1520947046428 p^{6} T^{10} + 203031496 p^{12} T^{12} + 16948 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
19 \( ( 1 - 21878 T^{2} + 133440805 T^{4} + 679703392870 T^{6} - 12186957975546872 T^{8} + 679703392870 p^{6} T^{10} + 133440805 p^{12} T^{12} - 21878 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
23 \( ( 1 - 54964 T^{2} + 1694889592 T^{4} - 33871122898172 T^{6} + 486652410292642670 T^{8} - 33871122898172 p^{6} T^{10} + 1694889592 p^{12} T^{12} - 54964 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
29 \( ( 1 - 102034 T^{2} + 6213617737 T^{4} - 245008733294114 T^{6} + 7064972052959541764 T^{8} - 245008733294114 p^{6} T^{10} + 6213617737 p^{12} T^{12} - 102034 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
31 \( ( 1 - 168680 T^{2} + 440701486 p T^{4} - 22418599073504 p T^{6} + 24523304532231348355 T^{8} - 22418599073504 p^{7} T^{10} + 440701486 p^{13} T^{12} - 168680 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
37 \( ( 1 - 598 T + 310003 T^{2} - 98555606 T^{3} + 26465731082 T^{4} - 98555606 p^{3} T^{5} + 310003 p^{6} T^{6} - 598 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
41 \( ( 1 + 148396 T^{2} + 19955386648 T^{4} + 1828971842691940 T^{6} + \)\(14\!\cdots\!50\)\( T^{8} + 1828971842691940 p^{6} T^{10} + 19955386648 p^{12} T^{12} + 148396 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
43 \( ( 1 - 82 T + 122239 T^{2} - 47448038 T^{3} + 6939760430 T^{4} - 47448038 p^{3} T^{5} + 122239 p^{6} T^{6} - 82 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
47 \( ( 1 + 358312 T^{2} + 37852166836 T^{4} - 1616224927676168 T^{6} - \)\(58\!\cdots\!34\)\( T^{8} - 1616224927676168 p^{6} T^{10} + 37852166836 p^{12} T^{12} + 358312 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
53 \( ( 1 - 506674 T^{2} + 170852400457 T^{4} - 38178976356454850 T^{6} + \)\(66\!\cdots\!00\)\( T^{8} - 38178976356454850 p^{6} T^{10} + 170852400457 p^{12} T^{12} - 506674 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
59 \( ( 1 + 399442 T^{2} + 69559586281 T^{4} - 2403193388843270 T^{6} - \)\(20\!\cdots\!44\)\( T^{8} - 2403193388843270 p^{6} T^{10} + 69559586281 p^{12} T^{12} + 399442 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
61 \( ( 1 - 1009100 T^{2} + 469105132744 T^{4} - 143096567459092868 T^{6} + \)\(34\!\cdots\!22\)\( T^{8} - 143096567459092868 p^{6} T^{10} + 469105132744 p^{12} T^{12} - 1009100 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
67 \( ( 1 + 154 T + 1033021 T^{2} + 1947442 p T^{3} + 445027403680 T^{4} + 1947442 p^{4} T^{5} + 1033021 p^{6} T^{6} + 154 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
71 \( ( 1 - 2546056 T^{2} + 2933736697204 T^{4} - 1997708473027518488 T^{6} + \)\(87\!\cdots\!74\)\( T^{8} - 1997708473027518488 p^{6} T^{10} + 2933736697204 p^{12} T^{12} - 2546056 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
73 \( ( 1 - 762938 T^{2} + 747028656433 T^{4} - 346998178646865506 T^{6} + \)\(17\!\cdots\!76\)\( T^{8} - 346998178646865506 p^{6} T^{10} + 747028656433 p^{12} T^{12} - 762938 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
79 \( ( 1 - 1088 T + 1452772 T^{2} - 1345149308 T^{3} + 1038291415081 T^{4} - 1345149308 p^{3} T^{5} + 1452772 p^{6} T^{6} - 1088 p^{9} T^{7} + p^{12} T^{8} )^{4} \)
83 \( ( 1 + 2632486 T^{2} + 3636170382349 T^{4} + 3366834686358200962 T^{6} + \)\(22\!\cdots\!84\)\( T^{8} + 3366834686358200962 p^{6} T^{10} + 3636170382349 p^{12} T^{12} + 2632486 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
89 \( ( 1 + 2548864 T^{2} + 2033941339900 T^{4} - 61305120471739520 T^{6} - \)\(81\!\cdots\!46\)\( T^{8} - 61305120471739520 p^{6} T^{10} + 2033941339900 p^{12} T^{12} + 2548864 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
97 \( ( 1 - 2978570 T^{2} + 5898756336049 T^{4} - 7901916867899068322 T^{6} + \)\(83\!\cdots\!24\)\( T^{8} - 7901916867899068322 p^{6} T^{10} + 5898756336049 p^{12} T^{12} - 2978570 p^{18} T^{14} + p^{24} T^{16} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.45185810030302730183816737198, −2.44605913056545081633274440601, −2.38291126327462303392227475828, −2.25113243780919595711980064934, −2.23812752142122884307037754602, −2.18880142974107168518900333316, −2.17061358834355910707038972717, −2.06996749742623482464346176783, −1.94708686818213920740037993911, −1.67379280233725219534066308983, −1.57907564301414753598083149391, −1.51458313742876646140496860714, −1.50399390311150455003408642091, −1.40593536985928911933323742898, −1.37216262772317272089426712356, −1.03700922907362904621671928705, −1.01149850023005358664082557902, −0.916128477039177770364381563902, −0.75182704675420247217558092807, −0.71849247831558934727287399603, −0.60362025812336524623301746553, −0.44598510318722759026548179506, −0.35192754438797187297000799858, −0.30318481579374790167520254007, −0.22714681955871365266122683747, 0.22714681955871365266122683747, 0.30318481579374790167520254007, 0.35192754438797187297000799858, 0.44598510318722759026548179506, 0.60362025812336524623301746553, 0.71849247831558934727287399603, 0.75182704675420247217558092807, 0.916128477039177770364381563902, 1.01149850023005358664082557902, 1.03700922907362904621671928705, 1.37216262772317272089426712356, 1.40593536985928911933323742898, 1.50399390311150455003408642091, 1.51458313742876646140496860714, 1.57907564301414753598083149391, 1.67379280233725219534066308983, 1.94708686818213920740037993911, 2.06996749742623482464346176783, 2.17061358834355910707038972717, 2.18880142974107168518900333316, 2.23812752142122884307037754602, 2.25113243780919595711980064934, 2.38291126327462303392227475828, 2.44605913056545081633274440601, 2.45185810030302730183816737198

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

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