Properties

Label 12-700e6-1.1-c5e6-0-1
Degree $12$
Conductor $1.176\times 10^{17}$
Sign $1$
Analytic cond. $2.00240\times 10^{12}$
Root an. cond. $10.5956$
Motivic weight $5$
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  + 448·9-s − 28·11-s − 4.65e3·19-s − 8.18e3·29-s + 1.17e4·31-s + 2.29e4·41-s − 7.20e3·49-s − 2.47e4·59-s + 5.43e4·61-s + 1.63e5·71-s + 3.18e4·79-s + 5.01e4·81-s + 1.91e5·89-s − 1.25e4·99-s − 4.89e4·101-s + 3.00e5·109-s − 6.71e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.37e6·169-s + ⋯
L(s)  = 1  + 1.84·9-s − 0.0697·11-s − 2.95·19-s − 1.80·29-s + 2.20·31-s + 2.12·41-s − 3/7·49-s − 0.926·59-s + 1.87·61-s + 3.86·71-s + 0.574·79-s + 0.849·81-s + 2.56·89-s − 0.128·99-s − 0.477·101-s + 2.42·109-s − 4.16·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 3.70·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 5^{12} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(2.00240\times 10^{12}\)
Root analytic conductor: \(10.5956\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 5^{12} \cdot 7^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(13.22700564\)
\(L(\frac12)\) \(\approx\) \(13.22700564\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( ( 1 + p^{4} T^{2} )^{3} \)
good3 \( 1 - 448 T^{2} + 150544 T^{4} - 5371766 p^{2} T^{6} + 150544 p^{10} T^{8} - 448 p^{20} T^{10} + p^{30} T^{12} \)
11 \( ( 1 + 14 T + 335922 T^{2} + 16946168 T^{3} + 335922 p^{5} T^{4} + 14 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
13 \( 1 - 1377000 T^{2} + 985345030872 T^{4} - 454375215957091750 T^{6} + 985345030872 p^{10} T^{8} - 1377000 p^{20} T^{10} + p^{30} T^{12} \)
17 \( 1 - 6850832 T^{2} + 21483625035184 T^{4} - 39031818380942816926 T^{6} + 21483625035184 p^{10} T^{8} - 6850832 p^{20} T^{10} + p^{30} T^{12} \)
19 \( ( 1 + 2328 T + 2524317 T^{2} + 99195680 T^{3} + 2524317 p^{5} T^{4} + 2328 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
23 \( 1 - 29109314 T^{2} + 395919058645231 T^{4} - \)\(32\!\cdots\!92\)\( T^{6} + 395919058645231 p^{10} T^{8} - 29109314 p^{20} T^{10} + p^{30} T^{12} \)
29 \( ( 1 + 4092 T + 44125116 T^{2} + 150282374694 T^{3} + 44125116 p^{5} T^{4} + 4092 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
31 \( ( 1 - 5888 T + 1430819 p T^{2} - 125334777344 T^{3} + 1430819 p^{6} T^{4} - 5888 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
37 \( 1 - 368704338 T^{2} + 59601431604256023 T^{4} - \)\(53\!\cdots\!84\)\( T^{6} + 59601431604256023 p^{10} T^{8} - 368704338 p^{20} T^{10} + p^{30} T^{12} \)
41 \( ( 1 - 11450 T + 303190107 T^{2} - 2174527049300 T^{3} + 303190107 p^{5} T^{4} - 11450 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
43 \( 1 - 512019274 T^{2} + 134589090211502951 T^{4} - \)\(23\!\cdots\!72\)\( T^{6} + 134589090211502951 p^{10} T^{8} - 512019274 p^{20} T^{10} + p^{30} T^{12} \)
47 \( 1 - 1161821288 T^{2} + 601652635689134632 T^{4} - \)\(17\!\cdots\!74\)\( T^{6} + 601652635689134632 p^{10} T^{8} - 1161821288 p^{20} T^{10} + p^{30} T^{12} \)
53 \( 1 - 2129732922 T^{2} + 2025868835335773543 T^{4} - \)\(10\!\cdots\!56\)\( T^{6} + 2025868835335773543 p^{10} T^{8} - 2129732922 p^{20} T^{10} + p^{30} T^{12} \)
59 \( ( 1 + 12388 T + 210415569 T^{2} - 23463075596456 T^{3} + 210415569 p^{5} T^{4} + 12388 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
61 \( ( 1 - 27182 T + 2359167815 T^{2} - 46085588664044 T^{3} + 2359167815 p^{5} T^{4} - 27182 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
67 \( 1 - 3129565474 T^{2} + 6761164680808370087 T^{4} - \)\(11\!\cdots\!52\)\( T^{6} + 6761164680808370087 p^{10} T^{8} - 3129565474 p^{20} T^{10} + p^{30} T^{12} \)
71 \( ( 1 - 81992 T + 4847939253 T^{2} - 182439241075184 T^{3} + 4847939253 p^{5} T^{4} - 81992 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
73 \( 1 + 1380006774 T^{2} - 2390193734302483713 T^{4} - \)\(19\!\cdots\!48\)\( T^{6} - 2390193734302483713 p^{10} T^{8} + 1380006774 p^{20} T^{10} + p^{30} T^{12} \)
79 \( ( 1 - 15926 T + 766889110 T^{2} + 176082125424572 T^{3} + 766889110 p^{5} T^{4} - 15926 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
83 \( 1 - 17184595170 T^{2} + \)\(13\!\cdots\!35\)\( T^{4} - \)\(67\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!35\)\( p^{10} T^{8} - 17184595170 p^{20} T^{10} + p^{30} T^{12} \)
89 \( ( 1 - 95710 T + 14224701483 T^{2} - 763443390521980 T^{3} + 14224701483 p^{5} T^{4} - 95710 p^{10} T^{5} + p^{15} T^{6} )^{2} \)
97 \( 1 - 23347801920 T^{2} + \)\(15\!\cdots\!72\)\( T^{4} - \)\(48\!\cdots\!10\)\( T^{6} + \)\(15\!\cdots\!72\)\( p^{10} T^{8} - 23347801920 p^{20} T^{10} + p^{30} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.75620247233483929461578110861, −4.39115726469236383750177947172, −4.21732467448871300950191923791, −4.21640115367422201315069247247, −4.20926557938158378355747168224, −4.04472661761417354231616865350, −4.00893469577725371635916367787, −3.51771690953161198621399785381, −3.35915323752538163270186320249, −3.07616928242842098758414996738, −3.05798328581133508057363590606, −2.90234602391144496448051167962, −2.60631501476762742201909060390, −2.22978328907376025874699502392, −2.09695042894755465682965997961, −1.92055613579485634676280672447, −1.82375153957687338823949971767, −1.81570999956989199154071814285, −1.71484268156701881966570971454, −0.935261937219353378877836082902, −0.849140714095480252219229620931, −0.77230428127230157803668292140, −0.72911885323636821140721839518, −0.46066245388220388121517235430, −0.20076036671580785435009989123, 0.20076036671580785435009989123, 0.46066245388220388121517235430, 0.72911885323636821140721839518, 0.77230428127230157803668292140, 0.849140714095480252219229620931, 0.935261937219353378877836082902, 1.71484268156701881966570971454, 1.81570999956989199154071814285, 1.82375153957687338823949971767, 1.92055613579485634676280672447, 2.09695042894755465682965997961, 2.22978328907376025874699502392, 2.60631501476762742201909060390, 2.90234602391144496448051167962, 3.05798328581133508057363590606, 3.07616928242842098758414996738, 3.35915323752538163270186320249, 3.51771690953161198621399785381, 4.00893469577725371635916367787, 4.04472661761417354231616865350, 4.20926557938158378355747168224, 4.21640115367422201315069247247, 4.21732467448871300950191923791, 4.39115726469236383750177947172, 4.75620247233483929461578110861

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

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