Properties

Label 32-33e32-1.1-c2e16-0-2
Degree $32$
Conductor $3.912\times 10^{48}$
Sign $1$
Analytic cond. $3.61249\times 10^{23}$
Root an. cond. $5.44730$
Motivic weight $2$
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  + 10·4-s + 67·16-s − 192·25-s − 80·31-s − 280·37-s + 610·49-s + 390·64-s + 300·67-s − 720·97-s − 1.92e3·100-s − 460·103-s − 800·124-s + 127-s + 131-s + 137-s + 139-s − 2.80e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.06e3·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 5/2·4-s + 4.18·16-s − 7.67·25-s − 2.58·31-s − 7.56·37-s + 12.4·49-s + 6.09·64-s + 4.47·67-s − 7.42·97-s − 19.1·100-s − 4.46·103-s − 6.45·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 18.9·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.27·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

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

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 11^{32}\)
Sign: $1$
Analytic conductor: \(3.61249\times 10^{23}\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 11^{32} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.464391775\)
\(L(\frac12)\) \(\approx\) \(3.464391775\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( ( 1 - 5 T^{2} + p^{2} T^{4} - 5 T^{6} + 161 T^{8} - 5 p^{4} T^{10} + p^{10} T^{12} - 5 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
5 \( ( 1 + 96 T^{2} + 4821 T^{4} + 164996 T^{6} + 4485981 T^{8} + 164996 p^{4} T^{10} + 4821 p^{8} T^{12} + 96 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
7 \( ( 1 - 305 T^{2} + 43654 T^{4} - 3833660 T^{6} + 226313501 T^{8} - 3833660 p^{4} T^{10} + 43654 p^{8} T^{12} - 305 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
13 \( ( 1 - 530 T^{2} + 165229 T^{4} - 35998190 T^{6} + 6655301396 T^{8} - 35998190 p^{4} T^{10} + 165229 p^{8} T^{12} - 530 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
17 \( ( 1 - 1135 T^{2} + 572919 T^{4} - 189780845 T^{6} + 54860904536 T^{8} - 189780845 p^{4} T^{10} + 572919 p^{8} T^{12} - 1135 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( ( 1 - 1343 T^{2} + 1000303 T^{4} - 510386801 T^{6} + 205804406480 T^{8} - 510386801 p^{4} T^{10} + 1000303 p^{8} T^{12} - 1343 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( ( 1 + 2802 T^{2} + 3878493 T^{4} + 3449803874 T^{6} + 2153442697260 T^{8} + 3449803874 p^{4} T^{10} + 3878493 p^{8} T^{12} + 2802 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
29 \( ( 1 - 2933 T^{2} + 4871983 T^{4} - 5832960491 T^{6} + 5544825136880 T^{8} - 5832960491 p^{4} T^{10} + 4871983 p^{8} T^{12} - 2933 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
31 \( ( 1 + 20 T + 1799 T^{2} + 43470 T^{3} + 2202931 T^{4} + 43470 p^{2} T^{5} + 1799 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
37 \( ( 1 + 70 T + 7011 T^{2} + 297060 T^{3} + 15426016 T^{4} + 297060 p^{2} T^{5} + 7011 p^{4} T^{6} + 70 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
41 \( ( 1 - 8038 T^{2} + 31956273 T^{4} - 83615361746 T^{6} + 161234535222620 T^{8} - 83615361746 p^{4} T^{10} + 31956273 p^{8} T^{12} - 8038 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
43 \( ( 1 - 9690 T^{2} + 47757189 T^{4} - 151131108070 T^{6} + 331901498261796 T^{8} - 151131108070 p^{4} T^{10} + 47757189 p^{8} T^{12} - 9690 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
47 \( ( 1 + 211 p T^{2} + 51498963 T^{4} + 178862393539 T^{6} + 454854480905960 T^{8} + 178862393539 p^{4} T^{10} + 51498963 p^{8} T^{12} + 211 p^{13} T^{14} + p^{16} T^{16} )^{2} \)
53 \( ( 1 + 6692 T^{2} + 40641113 T^{4} + 149532077364 T^{6} + 501716120871085 T^{8} + 149532077364 p^{4} T^{10} + 40641113 p^{8} T^{12} + 6692 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
59 \( ( 1 + 5628 T^{2} + 46132353 T^{4} + 151021447556 T^{6} + 755040081151485 T^{8} + 151021447556 p^{4} T^{10} + 46132353 p^{8} T^{12} + 5628 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
61 \( ( 1 - 14903 T^{2} + 115496143 T^{4} - 653200713881 T^{6} + 2818153490361680 T^{8} - 653200713881 p^{4} T^{10} + 115496143 p^{8} T^{12} - 14903 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
67 \( ( 1 - 75 T + 14301 T^{2} - 746575 T^{3} + 84609756 T^{4} - 746575 p^{2} T^{5} + 14301 p^{4} T^{6} - 75 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
71 \( ( 1 + 18773 T^{2} + 181913403 T^{4} + 1194030778411 T^{6} + 6439237809345560 T^{8} + 1194030778411 p^{4} T^{10} + 181913403 p^{8} T^{12} + 18773 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
73 \( ( 1 - 12415 T^{2} + 118927239 T^{4} - 869118101585 T^{6} + 5147586794154656 T^{8} - 869118101585 p^{4} T^{10} + 118927239 p^{8} T^{12} - 12415 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
79 \( ( 1 - 10193 T^{2} + 71196118 T^{4} - 7999979804 p T^{6} + 4700630217645605 T^{8} - 7999979804 p^{5} T^{10} + 71196118 p^{8} T^{12} - 10193 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
83 \( ( 1 - 19675 T^{2} + 282871194 T^{4} - 2855828501480 T^{6} + 22309213542340361 T^{8} - 2855828501480 p^{4} T^{10} + 282871194 p^{8} T^{12} - 19675 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
89 \( ( 1 + 34122 T^{2} + 499213893 T^{4} + 4369414395194 T^{6} + 32652563534737980 T^{8} + 4369414395194 p^{4} T^{10} + 499213893 p^{8} T^{12} + 34122 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
97 \( ( 1 + 180 T + 30331 T^{2} + 3393570 T^{3} + 348656791 T^{4} + 3393570 p^{2} T^{5} + 30331 p^{4} T^{6} + 180 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
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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

−2.26596586867059833599316626574, −2.23589393715988370801958539198, −2.22919937694997877859887980261, −2.09821244185421128075783828980, −1.99847802048800339306246234804, −1.91446199670971839116552992316, −1.87789973683575616287633332569, −1.79700337375762258415070618505, −1.76070274946302168668505793173, −1.55071406272815548916259603342, −1.51402130110666098654721895079, −1.45675727267353161009417842913, −1.41669996466522412333445729508, −1.37938334323965251800369334542, −1.15226165942045760316799473211, −1.11925602765433851493511526714, −0.924373455205116967640112021414, −0.77826498733628489827639639814, −0.74236240148438108040507305194, −0.63658261776292937522126434706, −0.50916670115686511173168558726, −0.33095594558828065018102198095, −0.25027971234986268961540958436, −0.23396189852075180132664169361, −0.05891822422001170395393895234, 0.05891822422001170395393895234, 0.23396189852075180132664169361, 0.25027971234986268961540958436, 0.33095594558828065018102198095, 0.50916670115686511173168558726, 0.63658261776292937522126434706, 0.74236240148438108040507305194, 0.77826498733628489827639639814, 0.924373455205116967640112021414, 1.11925602765433851493511526714, 1.15226165942045760316799473211, 1.37938334323965251800369334542, 1.41669996466522412333445729508, 1.45675727267353161009417842913, 1.51402130110666098654721895079, 1.55071406272815548916259603342, 1.76070274946302168668505793173, 1.79700337375762258415070618505, 1.87789973683575616287633332569, 1.91446199670971839116552992316, 1.99847802048800339306246234804, 2.09821244185421128075783828980, 2.22919937694997877859887980261, 2.23589393715988370801958539198, 2.26596586867059833599316626574

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

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