Properties

Label 32-3744e16-1.1-c1e16-0-0
Degree $32$
Conductor $1.491\times 10^{57}$
Sign $1$
Analytic cond. $4.07199\times 10^{23}$
Root an. cond. $5.46772$
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  − 40·25-s − 8·49-s − 32·79-s + 80·103-s − 96·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-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 + ⋯
L(s)  = 1  − 8·25-s − 8/7·49-s − 3.60·79-s + 7.88·103-s − 8.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{80} \cdot 3^{32} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(4.07199\times 10^{23}\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3744} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{80} \cdot 3^{32} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.02567959831\)
\(L(\frac12)\) \(\approx\) \(0.02567959831\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( ( 1 - 12 T^{2} + 54 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
good5 \( ( 1 + 2 p T^{2} + 14 p T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{4} \)
7 \( ( 1 + 2 T^{2} + 94 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
11 \( ( 1 + 24 T^{2} + 306 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
17 \( ( 1 + 8 T^{2} + 574 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
19 \( ( 1 + 42 T^{2} + 918 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
23 \( ( 1 + 12 T^{2} + 374 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
29 \( ( 1 - 88 T^{2} + 3598 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
31 \( ( 1 + 26 T^{2} + 1966 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
37 \( ( 1 + 44 T^{2} + 1222 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
41 \( ( 1 - 38 T^{2} + 78 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
43 \( ( 1 - 152 T^{2} + 9454 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
47 \( ( 1 - 104 T^{2} + 6802 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
53 \( ( 1 - 84 T^{2} + 3462 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
59 \( ( 1 + 136 T^{2} + 9586 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
61 \( ( 1 - 64 T^{2} + 6846 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
67 \( ( 1 + 114 T^{2} + 11622 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
71 \( ( 1 - 168 T^{2} + 14258 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
73 \( ( 1 - 92 T^{2} + 6294 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
79 \( ( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{8} \)
83 \( ( 1 + 232 T^{2} + 25234 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
89 \( ( 1 - 126 T^{2} + 18686 T^{4} - 126 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
97 \( ( 1 - 228 T^{2} + 28934 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
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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

−2.02760830744348645074468119266, −1.98913873278759726953898107865, −1.96720616184811833589236402002, −1.94654609711017424092161694398, −1.74436092226752312869185379652, −1.63021477994219260485274585972, −1.61674064223814282620648175016, −1.59296646648259450382841501096, −1.53583433469821506692681233970, −1.45893073111826272470711272288, −1.40674963321736047701679203458, −1.33233089371960447334404965914, −1.23129051431590356240145597679, −1.14625757179465024749321560761, −0.966660467320246193919145738151, −0.964018436896238674879896646475, −0.884477197281367230673619617988, −0.70602798363387383534832986836, −0.69990825632321635383724216256, −0.68102657421081151635870463418, −0.47983822295307819993214674586, −0.15436511485936360270871583126, −0.099028767953143538141606025447, −0.096095575243328341010274785497, −0.07500527494715604688295289499, 0.07500527494715604688295289499, 0.096095575243328341010274785497, 0.099028767953143538141606025447, 0.15436511485936360270871583126, 0.47983822295307819993214674586, 0.68102657421081151635870463418, 0.69990825632321635383724216256, 0.70602798363387383534832986836, 0.884477197281367230673619617988, 0.964018436896238674879896646475, 0.966660467320246193919145738151, 1.14625757179465024749321560761, 1.23129051431590356240145597679, 1.33233089371960447334404965914, 1.40674963321736047701679203458, 1.45893073111826272470711272288, 1.53583433469821506692681233970, 1.59296646648259450382841501096, 1.61674064223814282620648175016, 1.63021477994219260485274585972, 1.74436092226752312869185379652, 1.94654609711017424092161694398, 1.96720616184811833589236402002, 1.98913873278759726953898107865, 2.02760830744348645074468119266

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

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