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Properties

Label 12-1944e6-1.1-c0e6-0-5
Degree $12$
Conductor $5.397\times 10^{19}$
Sign $1$
Analytic cond. $0.833912$
Root an. cond. $0.984978$
Motivic weight $0$
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  + 8-s + 3·11-s + 3·41-s + 6·43-s + 3·59-s + 6·67-s + 3·88-s − 3·89-s − 3·97-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 8-s + 3·11-s + 3·41-s + 6·43-s + 3·59-s + 6·67-s + 3·88-s − 3·89-s − 3·97-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{30}\)
Sign: $1$
Analytic conductor: \(0.833912\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{30} ,\ ( \ : [0]^{6} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.834669484\)
\(L(\frac12)\) \(\approx\) \(2.834669484\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
3 \( 1 \)
good5 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
7 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
11 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
13 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
17 \( ( 1 - T^{3} + T^{6} )^{2} \)
19 \( ( 1 + T^{3} + T^{6} )^{2} \)
23 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
29 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
31 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
37 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
41 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
43 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
47 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
53 \( ( 1 - T )^{6}( 1 + T )^{6} \)
59 \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \)
61 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
67 \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
71 \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
73 \( ( 1 + T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
83 \( ( 1 - T^{3} + T^{6} )^{2} \)
89 \( ( 1 + T )^{6}( 1 - T + T^{2} )^{3} \)
97 \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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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

−5.04403419144909412456493830945, −4.81643388169252996442266453164, −4.64999547900774370734978514284, −4.52645776261526215737395790570, −4.07348333411546764142899657578, −4.06211706192589180833096946910, −4.06123165961833433305203749392, −4.02458419632516746309826095225, −3.96599278655358906830206531083, −3.93458893933579024584713088547, −3.59309863149272016895102466001, −3.30925062068482923950644190547, −3.23333530720737685700273102437, −2.77222240492862098589866283036, −2.71074827748417174778371657346, −2.49459858851301469656874134790, −2.32582936610298164701426781261, −2.32491104903267342083480341237, −2.18308043809103179629519509987, −1.92458444129011073624186614213, −1.29331507943810206805392541623, −1.24007254212845196666883464010, −1.09974060097411744207754349852, −1.05668458252014961602652953925, −0.927744116007820154604292628073, 0.927744116007820154604292628073, 1.05668458252014961602652953925, 1.09974060097411744207754349852, 1.24007254212845196666883464010, 1.29331507943810206805392541623, 1.92458444129011073624186614213, 2.18308043809103179629519509987, 2.32491104903267342083480341237, 2.32582936610298164701426781261, 2.49459858851301469656874134790, 2.71074827748417174778371657346, 2.77222240492862098589866283036, 3.23333530720737685700273102437, 3.30925062068482923950644190547, 3.59309863149272016895102466001, 3.93458893933579024584713088547, 3.96599278655358906830206531083, 4.02458419632516746309826095225, 4.06123165961833433305203749392, 4.06211706192589180833096946910, 4.07348333411546764142899657578, 4.52645776261526215737395790570, 4.64999547900774370734978514284, 4.81643388169252996442266453164, 5.04403419144909412456493830945

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

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