Properties

Label 12-5070e6-1.1-c1e6-0-6
Degree $12$
Conductor $1.698\times 10^{22}$
Sign $1$
Analytic cond. $4.40261\times 10^{9}$
Root an. cond. $6.36271$
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 − 3·4-s + 21·9-s + 18·12-s + 6·16-s − 6·17-s − 6·23-s − 3·25-s − 56·27-s − 20·29-s − 63·36-s − 38·43-s − 36·48-s + 28·49-s + 36·51-s + 38·53-s + 2·61-s − 10·64-s + 18·68-s + 36·69-s + 18·75-s + 30·79-s + 126·81-s + 120·87-s + 18·92-s + 9·100-s + 68·101-s + ⋯
L(s)  = 1  − 3.46·3-s − 3/2·4-s + 7·9-s + 5.19·12-s + 3/2·16-s − 1.45·17-s − 1.25·23-s − 3/5·25-s − 10.7·27-s − 3.71·29-s − 10.5·36-s − 5.79·43-s − 5.19·48-s + 4·49-s + 5.04·51-s + 5.21·53-s + 0.256·61-s − 5/4·64-s + 2.18·68-s + 4.33·69-s + 2.07·75-s + 3.37·79-s + 14·81-s + 12.8·87-s + 1.87·92-s + 9/10·100-s + 6.76·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4.40261\times 10^{9}\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8376863802\)
\(L(\frac12)\) \(\approx\) \(0.8376863802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
3 \( ( 1 + T )^{6} \)
5 \( ( 1 + T^{2} )^{3} \)
13 \( 1 \)
good7 \( 1 - 4 p T^{2} + 8 p^{2} T^{4} - 69 p^{2} T^{6} + 8 p^{4} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \)
11 \( 1 - 32 T^{2} + 492 T^{4} - 5573 T^{6} + 492 p^{2} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + 3 T + 47 T^{2} + 89 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 101 T^{2} + 4453 T^{4} - 110009 T^{6} + 4453 p^{2} T^{8} - 101 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 3 T + 51 T^{2} + 111 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( ( 1 + 10 T + 104 T^{2} + 539 T^{3} + 104 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 176 T^{2} + 13192 T^{4} - 539213 T^{6} + 13192 p^{2} T^{8} - 176 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 216 T^{2} + 19652 T^{4} - 964145 T^{6} + 19652 p^{2} T^{8} - 216 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 61 T^{2} + 3257 T^{4} - 171945 T^{6} + 3257 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 + 19 T + 219 T^{2} + 1747 T^{3} + 219 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 135 T^{2} + 11918 T^{4} - 649139 T^{6} + 11918 p^{2} T^{8} - 135 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 - 19 T + 221 T^{2} - 1931 T^{3} + 221 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 145 T^{2} + 5969 T^{4} - 100569 T^{6} + 5969 p^{2} T^{8} - 145 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 - T + 97 T^{2} - 373 T^{3} + 97 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 257 T^{2} + 33997 T^{4} - 2821217 T^{6} + 33997 p^{2} T^{8} - 257 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 197 T^{2} + 24417 T^{4} - 2019641 T^{6} + 24417 p^{2} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 + 71 T^{2} + 15761 T^{4} + 756591 T^{6} + 15761 p^{2} T^{8} + 71 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 15 T + 291 T^{2} - 2383 T^{3} + 291 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 388 T^{2} + 69852 T^{4} - 7373821 T^{6} + 69852 p^{2} T^{8} - 388 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 213 T^{2} + 14925 T^{4} - 642513 T^{6} + 14925 p^{2} T^{8} - 213 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 - 544 T^{2} + 126520 T^{4} - 16133233 T^{6} + 126520 p^{2} T^{8} - 544 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.33257777362928464317868344958, −4.06911403856038903403681610501, −3.92377408544048536593085586399, −3.83833589739774039399396316305, −3.79689068299856009326087889439, −3.65045148389312122381617627504, −3.63252999630653618805458313009, −3.35504845592714652108366990047, −3.32603190612409698154885624543, −2.99443411268074903542806114506, −2.80257865950877720234005857678, −2.43002090579720211344102340736, −2.33707369050268072767400543576, −2.17955412239119229068525332489, −1.94618358620296586297386764952, −1.89392803709015107334140406139, −1.82666995166732522605061441393, −1.71894822255597981223504137115, −1.43912050338735408733962110980, −0.995255823029770384440901979685, −0.854798684823446662482882527467, −0.68389633364134249059480563018, −0.45154405976837365496967814657, −0.38339072863749065379971307672, −0.31511429372794065955509428860, 0.31511429372794065955509428860, 0.38339072863749065379971307672, 0.45154405976837365496967814657, 0.68389633364134249059480563018, 0.854798684823446662482882527467, 0.995255823029770384440901979685, 1.43912050338735408733962110980, 1.71894822255597981223504137115, 1.82666995166732522605061441393, 1.89392803709015107334140406139, 1.94618358620296586297386764952, 2.17955412239119229068525332489, 2.33707369050268072767400543576, 2.43002090579720211344102340736, 2.80257865950877720234005857678, 2.99443411268074903542806114506, 3.32603190612409698154885624543, 3.35504845592714652108366990047, 3.63252999630653618805458313009, 3.65045148389312122381617627504, 3.79689068299856009326087889439, 3.83833589739774039399396316305, 3.92377408544048536593085586399, 4.06911403856038903403681610501, 4.33257777362928464317868344958

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

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