Properties

Label 12-462e6-1.1-c1e6-0-0
Degree $12$
Conductor $9.724\times 10^{15}$
Sign $1$
Analytic cond. $2520.66$
Root an. cond. $1.92070$
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  − 3·2-s − 3·3-s + 3·4-s + 9·6-s + 2·8-s + 3·9-s − 3·11-s − 9·12-s − 9·16-s − 3·17-s − 9·18-s + 3·19-s + 9·22-s − 9·23-s − 6·24-s + 6·25-s + 2·27-s + 18·29-s − 6·31-s + 9·32-s + 9·33-s + 9·34-s + 9·36-s − 9·37-s − 9·38-s + 24·41-s + 18·43-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 3/2·4-s + 3.67·6-s + 0.707·8-s + 9-s − 0.904·11-s − 2.59·12-s − 9/4·16-s − 0.727·17-s − 2.12·18-s + 0.688·19-s + 1.91·22-s − 1.87·23-s − 1.22·24-s + 6/5·25-s + 0.384·27-s + 3.34·29-s − 1.07·31-s + 1.59·32-s + 1.56·33-s + 1.54·34-s + 3/2·36-s − 1.47·37-s − 1.45·38-s + 3.74·41-s + 2.74·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(2520.66\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{462} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3746191558\)
\(L(\frac12)\) \(\approx\) \(0.3746191558\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T + T^{2} )^{3} \)
3 \( ( 1 + T + T^{2} )^{3} \)
7 \( 1 + 12 T^{2} - 4 T^{3} + 12 p T^{4} + p^{3} T^{6} \)
11 \( ( 1 + T + T^{2} )^{3} \)
good5 \( 1 - 6 T^{2} + 8 T^{3} + 6 T^{4} - 24 T^{5} + 86 T^{6} - 24 p T^{7} + 6 p^{2} T^{8} + 8 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 3 T^{2} + 32 T^{3} + 3 p T^{4} + p^{3} T^{6} )^{2} \)
17 \( ( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \)
19 \( 1 - 3 T - 12 T^{2} + 237 T^{3} - 408 T^{4} - 1839 T^{5} + 24590 T^{6} - 1839 p T^{7} - 408 p^{2} T^{8} + 237 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 9 T + 9 T^{2} - 6 p T^{3} - 9 T^{4} + 3249 T^{5} + 16702 T^{6} + 3249 p T^{7} - 9 p^{2} T^{8} - 6 p^{4} T^{9} + 9 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 9 T + 3 p T^{2} - 430 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 6 T + 39 T^{2} + 262 T^{3} + 942 T^{4} + 7302 T^{5} + 44307 T^{6} + 7302 p T^{7} + 942 p^{2} T^{8} + 262 p^{3} T^{9} + 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T - 18 T^{2} - 113 T^{3} + 1620 T^{4} - 3915 T^{5} - 120216 T^{6} - 3915 p T^{7} + 1620 p^{2} T^{8} - 113 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 12 T + 144 T^{2} - 22 p T^{3} + 144 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 9 T + 117 T^{2} - 610 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 15 T + 33 T^{2} - 98 T^{3} + 6975 T^{4} - 29487 T^{5} - 87106 T^{6} - 29487 p T^{7} + 6975 p^{2} T^{8} - 98 p^{3} T^{9} + 33 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 123 T^{2} + 64 T^{3} + 8610 T^{4} - 3936 T^{5} - 503059 T^{6} - 3936 p T^{7} + 8610 p^{2} T^{8} + 64 p^{3} T^{9} - 123 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 + 9 T + 141 T^{3} - 864 T^{4} - 30879 T^{5} - 171866 T^{6} - 30879 p T^{7} - 864 p^{2} T^{8} + 141 p^{3} T^{9} + 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 18 T^{2} + 688 T^{3} - 3414 T^{4} - 10122 T^{5} + 398166 T^{6} - 10122 p T^{7} - 3414 p^{2} T^{8} + 688 p^{3} T^{9} - 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T - 150 T^{2} - 324 T^{3} + 17814 T^{4} + 11310 T^{5} - 1359610 T^{6} + 11310 p T^{7} + 17814 p^{2} T^{8} - 324 p^{3} T^{9} - 150 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 3 T + 189 T^{2} - 362 T^{3} + 189 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 123 T^{2} - 384 T^{3} + 6150 T^{4} + 23616 T^{5} - 289519 T^{6} + 23616 p T^{7} + 6150 p^{2} T^{8} - 384 p^{3} T^{9} - 123 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 + 12 T - 60 T^{2} - 608 T^{3} + 7164 T^{4} - 180 T^{5} - 865506 T^{6} - 180 p T^{7} + 7164 p^{2} T^{8} - 608 p^{3} T^{9} - 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 + 6 T + 138 T^{2} + 1184 T^{3} + 138 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 36 T + 633 T^{2} - 8396 T^{3} + 102066 T^{4} - 1165620 T^{5} + 11868377 T^{6} - 1165620 p T^{7} + 102066 p^{2} T^{8} - 8396 p^{3} T^{9} + 633 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 9 T + 210 T^{2} - 1753 T^{3} + 210 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−6.06595842570451276506128523729, −5.87088351832905122411404056090, −5.64111985714492784038539891908, −5.49723374202152058128231518776, −5.29357517138920473619112426587, −5.15885838274558759677279372969, −5.05263811720458959983837046862, −4.54699400551725565503628389554, −4.52377027537023405314432433465, −4.48460393377776174844978349390, −4.14329712253266377842133933876, −4.10957094732783205129893072402, −4.02489999895463392542565645755, −3.33158883486157952311631093762, −3.21336850420293364127955590820, −2.95982745103062797297962047565, −2.89214629357818907268672379455, −2.46471176577586186006788628156, −2.18584371259481185497844531597, −2.08791356632927779332887819520, −1.70601479371730345511682395519, −1.28662519061894609373349649413, −0.71633729671301642371327671941, −0.70621898919584185543757569775, −0.53719408515018209833302588350, 0.53719408515018209833302588350, 0.70621898919584185543757569775, 0.71633729671301642371327671941, 1.28662519061894609373349649413, 1.70601479371730345511682395519, 2.08791356632927779332887819520, 2.18584371259481185497844531597, 2.46471176577586186006788628156, 2.89214629357818907268672379455, 2.95982745103062797297962047565, 3.21336850420293364127955590820, 3.33158883486157952311631093762, 4.02489999895463392542565645755, 4.10957094732783205129893072402, 4.14329712253266377842133933876, 4.48460393377776174844978349390, 4.52377027537023405314432433465, 4.54699400551725565503628389554, 5.05263811720458959983837046862, 5.15885838274558759677279372969, 5.29357517138920473619112426587, 5.49723374202152058128231518776, 5.64111985714492784038539891908, 5.87088351832905122411404056090, 6.06595842570451276506128523729

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

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