Properties

Label 12-177e6-1.1-c5e6-0-0
Degree $12$
Conductor $3.075\times 10^{13}$
Sign $1$
Analytic cond. $5.23362\times 10^{8}$
Root an. cond. $5.32803$
Motivic weight $5$
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  − 192·4-s + 2.15e4·16-s − 3.98e3·27-s − 1.83e6·64-s + 7.64e5·108-s − 9.66e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.22e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 6·4-s + 21·16-s − 1.05·27-s − 56·64-s + 6.30·108-s − 6·121-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 + 6·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s + 1.28e−6·227-s + 1.26e−6·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 59^{6}\)
Sign: $1$
Analytic conductor: \(5.23362\times 10^{8}\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 59^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(1.297069650\)
\(L(\frac12)\) \(\approx\) \(1.297069650\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3983 T^{3} + p^{15} T^{6} \)
59 \( ( 1 + p^{5} T^{2} )^{3} \)
good2 \( ( 1 + p^{5} T^{2} )^{6} \)
5 \( ( 1 - 63399 T^{3} + p^{15} T^{6} )( 1 + 63399 T^{3} + p^{15} T^{6} ) \)
7 \( ( 1 + 4256881 T^{3} + p^{15} T^{6} )^{2} \)
11 \( ( 1 + p^{5} T^{2} )^{6} \)
13 \( ( 1 - p^{5} T^{2} )^{6} \)
17 \( ( 1 - 2283 T + p^{5} T^{2} )^{3}( 1 + 2283 T + p^{5} T^{2} )^{3} \)
19 \( ( 1 + 5936692111 T^{3} + p^{15} T^{6} )^{2} \)
23 \( ( 1 + p^{5} T^{2} )^{6} \)
29 \( ( 1 - 84326289339 T^{3} + p^{15} T^{6} )( 1 + 84326289339 T^{3} + p^{15} T^{6} ) \)
31 \( ( 1 - p^{5} T^{2} )^{6} \)
37 \( ( 1 - p^{5} T^{2} )^{6} \)
41 \( ( 1 - 2487402070125 T^{3} + p^{15} T^{6} )( 1 + 2487402070125 T^{3} + p^{15} T^{6} ) \)
43 \( ( 1 - p^{5} T^{2} )^{6} \)
47 \( ( 1 + p^{5} T^{2} )^{6} \)
53 \( ( 1 - 13949582277867 T^{3} + p^{15} T^{6} )( 1 + 13949582277867 T^{3} + p^{15} T^{6} ) \)
61 \( ( 1 - p^{5} T^{2} )^{6} \)
67 \( ( 1 - p^{5} T^{2} )^{6} \)
71 \( ( 1 - 60675 T + p^{5} T^{2} )^{3}( 1 + 60675 T + p^{5} T^{2} )^{3} \)
73 \( ( 1 - p^{5} T^{2} )^{6} \)
79 \( ( 1 - 14391730713575 T^{3} + p^{15} T^{6} )^{2} \)
83 \( ( 1 + p^{5} T^{2} )^{6} \)
89 \( ( 1 + p^{5} T^{2} )^{6} \)
97 \( ( 1 - p^{5} T^{2} )^{6} \)
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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

−5.73533240695052914924884328931, −5.60940791421029063036682178827, −5.50996238274063698729212169954, −5.43893393923653429228546132682, −5.17662213856628252579482979317, −5.02722651034342416519027942209, −4.69371959999365124516558447782, −4.57314034280603070126093001327, −4.19522742716935936313883447974, −4.19166889560640835876225326701, −4.15745704638152719725346799770, −4.05747064257536914831777519991, −3.58366243921556658688691226213, −3.31704531904092638713121883553, −3.13955587279269194280119004764, −3.09290071687403850225322026677, −2.79199418591818840342852145919, −1.87304051097798361934430712277, −1.79881452584896459338560862693, −1.50399233522670350682208692355, −1.12485488483231663446804191111, −0.61780575586255061420920796622, −0.53778461490728733386474666220, −0.50366992135817344513325016886, −0.35775265686656443075088849696, 0.35775265686656443075088849696, 0.50366992135817344513325016886, 0.53778461490728733386474666220, 0.61780575586255061420920796622, 1.12485488483231663446804191111, 1.50399233522670350682208692355, 1.79881452584896459338560862693, 1.87304051097798361934430712277, 2.79199418591818840342852145919, 3.09290071687403850225322026677, 3.13955587279269194280119004764, 3.31704531904092638713121883553, 3.58366243921556658688691226213, 4.05747064257536914831777519991, 4.15745704638152719725346799770, 4.19166889560640835876225326701, 4.19522742716935936313883447974, 4.57314034280603070126093001327, 4.69371959999365124516558447782, 5.02722651034342416519027942209, 5.17662213856628252579482979317, 5.43893393923653429228546132682, 5.50996238274063698729212169954, 5.60940791421029063036682178827, 5.73533240695052914924884328931

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

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