Properties

Label 12-507e6-1.1-c1e6-0-5
Degree $12$
Conductor $1.698\times 10^{16}$
Sign $1$
Analytic cond. $4402.61$
Root an. cond. $2.01206$
Motivic weight $1$
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  + 6·3-s + 7·4-s + 21·9-s + 42·12-s + 23·16-s + 14·17-s − 4·23-s + 20·25-s + 56·27-s − 16·29-s + 147·36-s + 6·43-s + 138·48-s + 4·49-s + 84·51-s − 26·53-s − 26·61-s + 42·64-s + 98·68-s − 24·69-s + 120·75-s − 18·79-s + 126·81-s − 96·87-s − 28·92-s + 140·100-s − 4·103-s + ⋯
L(s)  = 1  + 3.46·3-s + 7/2·4-s + 7·9-s + 12.1·12-s + 23/4·16-s + 3.39·17-s − 0.834·23-s + 4·25-s + 10.7·27-s − 2.97·29-s + 49/2·36-s + 0.914·43-s + 19.9·48-s + 4/7·49-s + 11.7·51-s − 3.57·53-s − 3.32·61-s + 21/4·64-s + 11.8·68-s − 2.88·69-s + 13.8·75-s − 2.02·79-s + 14·81-s − 10.2·87-s − 2.91·92-s + 14·100-s − 0.394·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4402.61\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{507} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(55.83260340\)
\(L(\frac12)\) \(\approx\) \(55.83260340\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T )^{6} \)
13 \( 1 \)
good2 \( 1 - 7 T^{2} + 13 p T^{4} - 63 T^{6} + 13 p^{3} T^{8} - 7 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 - 4 p T^{2} + 192 T^{4} - 1169 T^{6} + 192 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \)
7 \( 1 - 4 T^{2} + 52 T^{4} - 181 T^{6} + 52 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 5 T^{2} + 117 T^{4} - 2177 T^{6} + 117 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 - 7 T + 65 T^{2} - 245 T^{3} + 65 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 13 T^{2} + 325 T^{4} - 3913 T^{6} + 325 p^{2} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 2 T + 26 T^{2} + 175 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 + 8 T + 92 T^{2} + 421 T^{3} + 92 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 76 T^{2} + 4456 T^{4} - 150973 T^{6} + 4456 p^{2} T^{8} - 76 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 152 T^{2} + 11596 T^{4} - 534953 T^{6} + 11596 p^{2} T^{8} - 152 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 241 T^{2} + 24401 T^{4} - 1328481 T^{6} + 24401 p^{2} T^{8} - 241 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 3 T + 104 T^{2} - 287 T^{3} + 104 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 39 T^{2} + 3585 T^{4} + 112983 T^{6} + 3585 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 + 13 T + 199 T^{2} + 1407 T^{3} + 199 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 158 T^{2} + 7527 T^{4} - 195812 T^{6} + 7527 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 + 13 T + 195 T^{2} + 1363 T^{3} + 195 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 333 T^{2} + 50297 T^{4} - 4342241 T^{6} + 50297 p^{2} T^{8} - 333 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 232 T^{2} + 32292 T^{4} - 2749741 T^{6} + 32292 p^{2} T^{8} - 232 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 316 T^{2} + 48500 T^{4} - 4463217 T^{6} + 48500 p^{2} T^{8} - 316 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 9 T + 215 T^{2} + 1253 T^{3} + 215 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 352 T^{2} + 59264 T^{4} - 6129693 T^{6} + 59264 p^{2} T^{8} - 352 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 493 T^{2} + 104273 T^{4} - 12160425 T^{6} + 104273 p^{2} T^{8} - 493 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 219 T^{2} + 40742 T^{4} - 4194431 T^{6} + 40742 p^{2} T^{8} - 219 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.94168052563671430389541916584, −5.69319865909817833229851154249, −5.59350790785620267174477527742, −5.57126674527757022850408423181, −5.24178115727852240199357477249, −4.84836584027712463498210183411, −4.63888975635074699043875745028, −4.60814935532054668690156607025, −4.51797539478318427931039286220, −3.92448575173771854660425022154, −3.86212634835838989340430360574, −3.67340077188047881825447126990, −3.47790932776250483432896865000, −3.16978837543777033337460658369, −3.05734912950708440012946254322, −3.02789763394512717093033262098, −2.82728474994628151505669253728, −2.64290923768552615752740368747, −2.60781302073818172050014928076, −2.01890488757516149850030633341, −1.86597989438377714915301595061, −1.83456533850164448457965167917, −1.44473582901668101392635924908, −1.30744597933420542194274042478, −1.06490786971069116226107588199, 1.06490786971069116226107588199, 1.30744597933420542194274042478, 1.44473582901668101392635924908, 1.83456533850164448457965167917, 1.86597989438377714915301595061, 2.01890488757516149850030633341, 2.60781302073818172050014928076, 2.64290923768552615752740368747, 2.82728474994628151505669253728, 3.02789763394512717093033262098, 3.05734912950708440012946254322, 3.16978837543777033337460658369, 3.47790932776250483432896865000, 3.67340077188047881825447126990, 3.86212634835838989340430360574, 3.92448575173771854660425022154, 4.51797539478318427931039286220, 4.60814935532054668690156607025, 4.63888975635074699043875745028, 4.84836584027712463498210183411, 5.24178115727852240199357477249, 5.57126674527757022850408423181, 5.59350790785620267174477527742, 5.69319865909817833229851154249, 5.94168052563671430389541916584

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

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