Properties

Degree 32
Conductor $ 2^{80} \cdot 3^{48} \cdot 7^{16} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·7-s + 24·25-s − 40·31-s + 136·49-s + 24·73-s − 24·79-s + 16·97-s − 24·103-s + 120·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 80·169-s + 173-s + 384·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 6.04·7-s + 24/5·25-s − 7.18·31-s + 19.4·49-s + 2.80·73-s − 2.70·79-s + 1.62·97-s − 2.36·103-s + 10.9·121-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 + 6.15·169-s + 0.0760·173-s + 29.0·175-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(32\)
\( N \)  =  \(2^{80} \cdot 3^{48} \cdot 7^{16}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(32,\ 2^{80} \cdot 3^{48} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )$
$L(1)$  $\approx$  $331.8853758$
$L(\frac12)$  $\approx$  $331.8853758$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 32. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 - T )^{16} \)
good5 \( ( 1 - 12 T^{2} + 54 T^{4} - 48 T^{6} - 469 T^{8} - 48 p^{2} T^{10} + 54 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
11 \( ( 1 - 60 T^{2} + 1734 T^{4} - 31920 T^{6} + 413051 T^{8} - 31920 p^{2} T^{10} + 1734 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
13 \( ( 1 - 40 T^{2} + 72 p T^{4} - 16040 T^{6} + 219326 T^{8} - 16040 p^{2} T^{10} + 72 p^{5} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
17 \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{8} \)
19 \( ( 1 - 66 T^{2} + 1791 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} )^{4} \)
23 \( ( 1 + 52 T^{2} + 2430 T^{4} + 71312 T^{6} + 1906499 T^{8} + 71312 p^{2} T^{10} + 2430 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 44 T^{2} + 960 T^{4} - 644 p T^{6} + 833774 T^{8} - 644 p^{3} T^{10} + 960 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 + 10 T + 114 T^{2} + 800 T^{3} + 5351 T^{4} + 800 p T^{5} + 114 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
37 \( ( 1 - 192 T^{2} + 17010 T^{4} - 960672 T^{6} + 40219139 T^{8} - 960672 p^{2} T^{10} + 17010 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
41 \( ( 1 + 220 T^{2} + 24294 T^{4} + 1713680 T^{6} + 83598011 T^{8} + 1713680 p^{2} T^{10} + 24294 p^{4} T^{12} + 220 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 + 32 T^{2} + 4920 T^{4} + 158992 T^{6} + 11619614 T^{8} + 158992 p^{2} T^{10} + 4920 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( ( 1 + 124 T^{2} + 7312 T^{4} + 372292 T^{6} + 18784750 T^{8} + 372292 p^{2} T^{10} + 7312 p^{4} T^{12} + 124 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 216 T^{2} + 24972 T^{4} - 2015208 T^{6} + 122805830 T^{8} - 2015208 p^{2} T^{10} + 24972 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( ( 1 - 380 T^{2} + 67344 T^{4} - 7257700 T^{6} + 519439886 T^{8} - 7257700 p^{2} T^{10} + 67344 p^{4} T^{12} - 380 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 304 T^{2} + 46620 T^{4} - 4694096 T^{6} + 336476774 T^{8} - 4694096 p^{2} T^{10} + 46620 p^{4} T^{12} - 304 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 - 232 T^{2} + 32520 T^{4} - 3258632 T^{6} + 244468094 T^{8} - 3258632 p^{2} T^{10} + 32520 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 396 T^{2} + 77350 T^{4} + 9537264 T^{6} + 808347579 T^{8} + 9537264 p^{2} T^{10} + 77350 p^{4} T^{12} + 396 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 6 T + 238 T^{2} - 990 T^{3} + 23766 T^{4} - 990 p T^{5} + 238 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
79 \( ( 1 + 6 T + 82 T^{2} + 18 T^{3} + 6630 T^{4} + 18 p T^{5} + 82 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
83 \( ( 1 - 252 T^{2} + 47280 T^{4} - 5629572 T^{6} + 552317774 T^{8} - 5629572 p^{2} T^{10} + 47280 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
89 \( ( 1 + 364 T^{2} + 76230 T^{4} + 10653296 T^{6} + 1102200539 T^{8} + 10653296 p^{2} T^{10} + 76230 p^{4} T^{12} + 364 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( ( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{8} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.86897003357184962558569056632, −1.81503362218473476574393463283, −1.70121632273132490629122292087, −1.64638275559828341836632454807, −1.61724384461978845933420917491, −1.59857031474942986376177477844, −1.57820048152324375141538590388, −1.52577281624139652726553959719, −1.46351704182576376838518671288, −1.28723452011757111973048910573, −1.23905429539520531939281755259, −1.19383714320830173237702930981, −1.15479111511585884856525419584, −1.00693515594459741845397051248, −0.975624234655695822772924480150, −0.876711664347888639587491810701, −0.845744576586133467966226848296, −0.73730701825298773828021694141, −0.70193043259938654259965434906, −0.62530514474568575639686401884, −0.50561342925499237006933202823, −0.34072835190528893378823779008, −0.27540053526496939001118057155, −0.24775036261246403502097418460, −0.20408788389810471514423356198, 0.20408788389810471514423356198, 0.24775036261246403502097418460, 0.27540053526496939001118057155, 0.34072835190528893378823779008, 0.50561342925499237006933202823, 0.62530514474568575639686401884, 0.70193043259938654259965434906, 0.73730701825298773828021694141, 0.845744576586133467966226848296, 0.876711664347888639587491810701, 0.975624234655695822772924480150, 1.00693515594459741845397051248, 1.15479111511585884856525419584, 1.19383714320830173237702930981, 1.23905429539520531939281755259, 1.28723452011757111973048910573, 1.46351704182576376838518671288, 1.52577281624139652726553959719, 1.57820048152324375141538590388, 1.59857031474942986376177477844, 1.61724384461978845933420917491, 1.64638275559828341836632454807, 1.70121632273132490629122292087, 1.81503362218473476574393463283, 1.86897003357184962558569056632

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

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