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Properties

Label 12-722e6-1.1-c1e6-0-5
Degree $12$
Conductor $1.417\times 10^{17}$
Sign $1$
Analytic cond. $36718.5$
Root an. cond. $2.40108$
Motivic weight $1$
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  + 3·3-s − 6·7-s + 8-s + 3·9-s − 6·11-s − 12·13-s − 12·17-s − 18·21-s − 12·23-s + 3·24-s − 27-s + 18·29-s − 6·31-s − 18·33-s + 12·37-s − 36·39-s − 3·41-s − 6·43-s + 30·47-s + 21·49-s − 36·51-s − 24·53-s − 6·56-s + 3·59-s + 6·61-s − 18·63-s + 9·67-s + ⋯
L(s)  = 1  + 1.73·3-s − 2.26·7-s + 0.353·8-s + 9-s − 1.80·11-s − 3.32·13-s − 2.91·17-s − 3.92·21-s − 2.50·23-s + 0.612·24-s − 0.192·27-s + 3.34·29-s − 1.07·31-s − 3.13·33-s + 1.97·37-s − 5.76·39-s − 0.468·41-s − 0.914·43-s + 4.37·47-s + 3·49-s − 5.04·51-s − 3.29·53-s − 0.801·56-s + 0.390·59-s + 0.768·61-s − 2.26·63-s + 1.09·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(36718.5\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3693111490\)
\(L(\frac12)\) \(\approx\) \(0.3693111490\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
19 \( 1 \)
good3 \( 1 - p T + 2 p T^{2} - 8 T^{3} + 7 p T^{4} - 17 p T^{5} + 109 T^{6} - 17 p^{2} T^{7} + 7 p^{3} T^{8} - 8 p^{3} T^{9} + 2 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 22 T^{3} + 359 T^{6} - 22 p^{3} T^{9} + p^{6} T^{12} \)
7 \( 1 + 6 T + 15 T^{2} + 6 T^{3} - 66 T^{4} - 30 p T^{5} - 565 T^{6} - 30 p^{2} T^{7} - 66 p^{2} T^{8} + 6 p^{3} T^{9} + 15 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T - 10 T^{3} + 222 T^{4} + 42 T^{5} - 3181 T^{6} + 42 p T^{7} + 222 p^{2} T^{8} - 10 p^{3} T^{9} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 12 T + 48 T^{2} - 14 T^{3} - 684 T^{4} - 1854 T^{5} - 2757 T^{6} - 1854 p T^{7} - 684 p^{2} T^{8} - 14 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 12 T + 108 T^{2} + 702 T^{3} + 3744 T^{4} + 17904 T^{5} + 75187 T^{6} + 17904 p T^{7} + 3744 p^{2} T^{8} + 702 p^{3} T^{9} + 108 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 12 T + 60 T^{2} + 152 T^{3} - 84 T^{4} - 4734 T^{5} - 34459 T^{6} - 4734 p T^{7} - 84 p^{2} T^{8} + 152 p^{3} T^{9} + 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 18 T + 144 T^{2} - 20 p T^{3} + 1440 T^{4} - 7542 T^{5} + 56483 T^{6} - 7542 p T^{7} + 1440 p^{2} T^{8} - 20 p^{4} T^{9} + 144 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 6 T - 33 T^{2} - 346 T^{3} + 342 T^{4} + 6318 T^{5} + 21795 T^{6} + 6318 p T^{7} + 342 p^{2} T^{8} - 346 p^{3} T^{9} - 33 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 - 6 T + 87 T^{2} - 308 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 3 T + 6 T^{2} + 8 T^{3} - 111 T^{4} - 9711 T^{5} - 83311 T^{6} - 9711 p T^{7} - 111 p^{2} T^{8} + 8 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 6 T + 42 T^{2} - 116 T^{3} + 1044 T^{4} - 6768 T^{5} + 34173 T^{6} - 6768 p T^{7} + 1044 p^{2} T^{8} - 116 p^{3} T^{9} + 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 30 T + 372 T^{2} - 2086 T^{3} - 2544 T^{4} + 153360 T^{5} - 1460047 T^{6} + 153360 p T^{7} - 2544 p^{2} T^{8} - 2086 p^{3} T^{9} + 372 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 276 T^{2} + 2152 T^{3} + 18132 T^{4} + 179262 T^{5} + 1523855 T^{6} + 179262 p T^{7} + 18132 p^{2} T^{8} + 2152 p^{3} T^{9} + 276 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 3 T + 54 T^{2} - 378 T^{3} + 5823 T^{4} - 681 p T^{5} + 259255 T^{6} - 681 p^{2} T^{7} + 5823 p^{2} T^{8} - 378 p^{3} T^{9} + 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 12 T^{2} - 586 T^{3} - 1008 T^{4} + 15732 T^{5} + 419787 T^{6} + 15732 p T^{7} - 1008 p^{2} T^{8} - 586 p^{3} T^{9} - 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T + 162 T^{2} - 1854 T^{3} + 22437 T^{4} - 180369 T^{5} + 1838609 T^{6} - 180369 p T^{7} + 22437 p^{2} T^{8} - 1854 p^{3} T^{9} + 162 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 18 T + 216 T^{2} - 2516 T^{3} + 26244 T^{4} - 221256 T^{5} + 1869317 T^{6} - 221256 p T^{7} + 26244 p^{2} T^{8} - 2516 p^{3} T^{9} + 216 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 30 T + 279 T^{2} - 231 T^{3} - 13599 T^{4} + 110865 T^{5} + 2572046 T^{6} + 110865 p T^{7} - 13599 p^{2} T^{8} - 231 p^{3} T^{9} + 279 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 6 T - 48 T^{2} - 1430 T^{3} - 8820 T^{4} + 56934 T^{5} + 1359837 T^{6} + 56934 p T^{7} - 8820 p^{2} T^{8} - 1430 p^{3} T^{9} - 48 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 6 T - 186 T^{2} - 558 T^{3} + 25188 T^{4} + 30012 T^{5} - 2301977 T^{6} + 30012 p T^{7} + 25188 p^{2} T^{8} - 558 p^{3} T^{9} - 186 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 36 T^{2} + 486 T^{3} - 1548 T^{4} + 34956 T^{5} + 586603 T^{6} + 34956 p T^{7} - 1548 p^{2} T^{8} + 486 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 3 T + 6 T^{2} + 8 T^{3} + 585 T^{4} + 29097 T^{5} - 744183 T^{6} + 29097 p T^{7} + 585 p^{2} T^{8} + 8 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.77485778081367932885985263654, −5.26380958699408767678311106320, −5.18509393298248171708799695934, −5.00523730353697710939814009630, −4.83827359827261362778886308781, −4.41718586429100400361218531273, −4.40031936905926588628834640541, −4.39991105498692192660518615128, −4.38574046394321029680102351357, −4.06148619852762989937823851184, −3.81732368892598222459900422812, −3.46565050364324814753891095123, −3.22154758886822810932585796141, −3.13639733658937900151648796607, −2.91171521220856281850089386641, −2.90270788121487204115695298818, −2.42095863596302507516877315094, −2.38474138721532130223687871770, −2.32288552666733535725052425305, −2.23300164400365829430632797086, −2.12664725050634724846688482427, −1.54473844178841513247247095117, −1.04098219479655844025625102162, −0.38923622748038967152067529497, −0.18148679762108120056524935735, 0.18148679762108120056524935735, 0.38923622748038967152067529497, 1.04098219479655844025625102162, 1.54473844178841513247247095117, 2.12664725050634724846688482427, 2.23300164400365829430632797086, 2.32288552666733535725052425305, 2.38474138721532130223687871770, 2.42095863596302507516877315094, 2.90270788121487204115695298818, 2.91171521220856281850089386641, 3.13639733658937900151648796607, 3.22154758886822810932585796141, 3.46565050364324814753891095123, 3.81732368892598222459900422812, 4.06148619852762989937823851184, 4.38574046394321029680102351357, 4.39991105498692192660518615128, 4.40031936905926588628834640541, 4.41718586429100400361218531273, 4.83827359827261362778886308781, 5.00523730353697710939814009630, 5.18509393298248171708799695934, 5.26380958699408767678311106320, 5.77485778081367932885985263654

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

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