Properties

Label 12-2205e6-1.1-c3e6-0-0
Degree $12$
Conductor $1.149\times 10^{20}$
Sign $1$
Analytic cond. $4.84895\times 10^{12}$
Root an. cond. $11.4061$
Motivic weight $3$
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  − 14·4-s + 30·5-s + 83·16-s + 44·17-s − 420·20-s + 525·25-s + 380·37-s + 612·41-s − 328·43-s + 120·47-s + 136·59-s − 556·64-s + 1.11e3·67-s − 616·68-s + 1.40e3·79-s + 2.49e3·80-s + 2.91e3·83-s + 1.32e3·85-s + 372·89-s − 7.35e3·100-s + 3.19e3·101-s − 684·109-s − 4.47e3·121-s + 7.00e3·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 7/4·4-s + 2.68·5-s + 1.29·16-s + 0.627·17-s − 4.69·20-s + 21/5·25-s + 1.68·37-s + 2.33·41-s − 1.16·43-s + 0.372·47-s + 0.300·59-s − 1.08·64-s + 2.02·67-s − 1.09·68-s + 1.99·79-s + 3.47·80-s + 3.85·83-s + 1.68·85-s + 0.443·89-s − 7.34·100-s + 3.14·101-s − 0.601·109-s − 3.36·121-s + 5.00·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 5^{6} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(4.84895\times 10^{12}\)
Root analytic conductor: \(11.4061\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2205} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(40.80720372\)
\(L(\frac12)\) \(\approx\) \(40.80720372\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( ( 1 - p T )^{6} \)
7 \( 1 \)
good2 \( 1 + 7 p T^{2} + 113 T^{4} + 61 p^{4} T^{6} + 113 p^{6} T^{8} + 7 p^{13} T^{10} + p^{18} T^{12} \)
11 \( 1 + 4474 T^{2} + 10690519 T^{4} + 16986506732 T^{6} + 10690519 p^{6} T^{8} + 4474 p^{12} T^{10} + p^{18} T^{12} \)
13 \( 1 + 6438 T^{2} + 27862663 T^{4} + 71083547284 T^{6} + 27862663 p^{6} T^{8} + 6438 p^{12} T^{10} + p^{18} T^{12} \)
17 \( ( 1 - 22 T + 8879 T^{2} + 428 T^{3} + 8879 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
19 \( 1 + 21978 T^{2} + 221297479 T^{4} + 1589592507244 T^{6} + 221297479 p^{6} T^{8} + 21978 p^{12} T^{10} + p^{18} T^{12} \)
23 \( 1 + 49178 T^{2} + 1213984799 T^{4} + 18399433839724 T^{6} + 1213984799 p^{6} T^{8} + 49178 p^{12} T^{10} + p^{18} T^{12} \)
29 \( 1 + 124478 T^{2} + 6905589367 T^{4} + 217717211232388 T^{6} + 6905589367 p^{6} T^{8} + 124478 p^{12} T^{10} + p^{18} T^{12} \)
31 \( 1 + 51250 T^{2} + 1551151871 T^{4} + 34382118836764 T^{6} + 1551151871 p^{6} T^{8} + 51250 p^{12} T^{10} + p^{18} T^{12} \)
37 \( ( 1 - 190 T + 99155 T^{2} - 19441940 T^{3} + 99155 p^{3} T^{4} - 190 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
41 \( ( 1 - 306 T + 158503 T^{2} - 26678652 T^{3} + 158503 p^{3} T^{4} - 306 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
43 \( ( 1 + 164 T + 238921 T^{2} + 25708696 T^{3} + 238921 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
47 \( ( 1 - 60 T + 110269 T^{2} + 5181240 T^{3} + 110269 p^{3} T^{4} - 60 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
53 \( 1 + 558230 T^{2} + 156874030999 T^{4} + 28183915930956148 T^{6} + 156874030999 p^{6} T^{8} + 558230 p^{12} T^{10} + p^{18} T^{12} \)
59 \( ( 1 - 68 T + 155897 T^{2} + 40876456 T^{3} + 155897 p^{3} T^{4} - 68 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
61 \( 1 + 807486 T^{2} + 291238309815 T^{4} + 72313154579372420 T^{6} + 291238309815 p^{6} T^{8} + 807486 p^{12} T^{10} + p^{18} T^{12} \)
67 \( ( 1 - 556 T + 262353 T^{2} + 43010104 T^{3} + 262353 p^{3} T^{4} - 556 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
71 \( 1 + 1993522 T^{2} + 1707171677167 T^{4} + 802969695744376412 T^{6} + 1707171677167 p^{6} T^{8} + 1993522 p^{12} T^{10} + p^{18} T^{12} \)
73 \( 1 + 361278 T^{2} - 19156046897 T^{4} - 93754135461399356 T^{6} - 19156046897 p^{6} T^{8} + 361278 p^{12} T^{10} + p^{18} T^{12} \)
79 \( ( 1 - 700 T + 1616525 T^{2} - 696310024 T^{3} + 1616525 p^{3} T^{4} - 700 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
83 \( ( 1 - 1456 T + 1885361 T^{2} - 1479178144 T^{3} + 1885361 p^{3} T^{4} - 1456 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
89 \( ( 1 - 186 T + 1188007 T^{2} - 166563468 T^{3} + 1188007 p^{3} T^{4} - 186 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
97 \( 1 + 3688174 T^{2} + 6610702894847 T^{4} + 7408695484892503204 T^{6} + 6610702894847 p^{6} T^{8} + 3688174 p^{12} T^{10} + p^{18} T^{12} \)
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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

−4.44885908598209440488038566098, −3.99628107289914328880987689160, −3.98540197073906517479608220829, −3.89820370139880582283818880750, −3.88409317242052641865032533267, −3.63959970941453307786083039591, −3.41136702956808148589381989914, −3.25099418534408643726125457154, −3.10675847474109729668896067031, −2.77817121043565244715548157976, −2.76752035661379695235967122454, −2.46625098880710676226692208646, −2.38711252473695870567400913308, −2.34337675290273105615178716197, −2.18164709992506867996752174575, −1.80425247986902215963825516790, −1.61049076730129647178726433636, −1.54008651071844961356401458186, −1.50934030570331313607083367000, −1.08700048755751358991927298345, −0.838726207795432649361489770264, −0.64507118477363418421099294959, −0.60495195319578403255336954771, −0.49161124504479331312284023424, −0.37438079540959718497981072365, 0.37438079540959718497981072365, 0.49161124504479331312284023424, 0.60495195319578403255336954771, 0.64507118477363418421099294959, 0.838726207795432649361489770264, 1.08700048755751358991927298345, 1.50934030570331313607083367000, 1.54008651071844961356401458186, 1.61049076730129647178726433636, 1.80425247986902215963825516790, 2.18164709992506867996752174575, 2.34337675290273105615178716197, 2.38711252473695870567400913308, 2.46625098880710676226692208646, 2.76752035661379695235967122454, 2.77817121043565244715548157976, 3.10675847474109729668896067031, 3.25099418534408643726125457154, 3.41136702956808148589381989914, 3.63959970941453307786083039591, 3.88409317242052641865032533267, 3.89820370139880582283818880750, 3.98540197073906517479608220829, 3.99628107289914328880987689160, 4.44885908598209440488038566098

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

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