Properties

Label 12-28e12-1.1-c3e6-0-1
Degree $12$
Conductor $2.322\times 10^{17}$
Sign $1$
Analytic cond. $9.79699\times 10^{9}$
Root an. cond. $6.80128$
Motivic weight $3$
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  − 14·3-s + 95·9-s + 286·19-s + 69·25-s − 410·27-s − 348·29-s + 410·31-s + 498·37-s + 150·47-s + 1.29e3·53-s − 4.00e3·57-s + 642·59-s − 966·75-s + 71·81-s − 24·83-s + 4.87e3·87-s − 5.74e3·93-s + 4.19e3·103-s − 342·109-s − 6.97e3·111-s − 1.86e3·113-s + 6.16e3·121-s + 127-s + 131-s + 137-s + 139-s − 2.10e3·141-s + ⋯
L(s)  = 1  − 2.69·3-s + 3.51·9-s + 3.45·19-s + 0.551·25-s − 2.92·27-s − 2.22·29-s + 2.37·31-s + 2.21·37-s + 0.465·47-s + 3.34·53-s − 9.30·57-s + 1.41·59-s − 1.48·75-s + 0.0973·81-s − 0.0317·83-s + 6.00·87-s − 6.40·93-s + 4.00·103-s − 0.300·109-s − 5.96·111-s − 1.54·113-s + 4.63·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 1.25·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(9.79699\times 10^{9}\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{784} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(5.813535764\)
\(L(\frac12)\) \(\approx\) \(5.813535764\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( ( 1 + 7 T + 26 T^{2} + 65 T^{3} + 26 p^{3} T^{4} + 7 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
5 \( 1 - 69 T^{2} + 7374 T^{4} - 3424993 T^{6} + 7374 p^{6} T^{8} - 69 p^{12} T^{10} + p^{18} T^{12} \)
11 \( 1 - 6165 T^{2} + 17148690 T^{4} - 28517484157 T^{6} + 17148690 p^{6} T^{8} - 6165 p^{12} T^{10} + p^{18} T^{12} \)
13 \( 1 - 606 p T^{2} + 33176199 T^{4} - 88766411348 T^{6} + 33176199 p^{6} T^{8} - 606 p^{13} T^{10} + p^{18} T^{12} \)
17 \( 1 - 117 T^{2} + 459126 T^{4} - 235328100577 T^{6} + 459126 p^{6} T^{8} - 117 p^{12} T^{10} + p^{18} T^{12} \)
19 \( ( 1 - 143 T + 26114 T^{2} - 1994945 T^{3} + 26114 p^{3} T^{4} - 143 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
23 \( 1 - 50325 T^{2} + 1246242090 T^{4} - 19019633111629 T^{6} + 1246242090 p^{6} T^{8} - 50325 p^{12} T^{10} + p^{18} T^{12} \)
29 \( ( 1 + 6 p T + 45663 T^{2} + 3864972 T^{3} + 45663 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} )^{2} \)
31 \( ( 1 - 205 T + 60302 T^{2} - 10681531 T^{3} + 60302 p^{3} T^{4} - 205 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
37 \( ( 1 - 249 T + 118434 T^{2} - 21412069 T^{3} + 118434 p^{3} T^{4} - 249 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
41 \( 1 - 4350 p T^{2} + 20420082255 T^{4} - 1736555393379940 T^{6} + 20420082255 p^{6} T^{8} - 4350 p^{13} T^{10} + p^{18} T^{12} \)
43 \( 1 - 225246 T^{2} + 31219429287 T^{4} - 3038313220434308 T^{6} + 31219429287 p^{6} T^{8} - 225246 p^{12} T^{10} + p^{18} T^{12} \)
47 \( ( 1 - 75 T + 276918 T^{2} - 15429621 T^{3} + 276918 p^{3} T^{4} - 75 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
53 \( ( 1 - 645 T + 330090 T^{2} - 130877241 T^{3} + 330090 p^{3} T^{4} - 645 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
59 \( ( 1 - 321 T + 646890 T^{2} - 132762111 T^{3} + 646890 p^{3} T^{4} - 321 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
61 \( 1 - 1004301 T^{2} + 482746298670 T^{4} - 138139905664149977 T^{6} + 482746298670 p^{6} T^{8} - 1004301 p^{12} T^{10} + p^{18} T^{12} \)
67 \( 1 - 710157 T^{2} + 400704146994 T^{4} - 132106559938154597 T^{6} + 400704146994 p^{6} T^{8} - 710157 p^{12} T^{10} + p^{18} T^{12} \)
71 \( 1 - 730518 T^{2} + 329494751679 T^{4} - 139370995079650996 T^{6} + 329494751679 p^{6} T^{8} - 730518 p^{12} T^{10} + p^{18} T^{12} \)
73 \( 1 - 925629 T^{2} + 643454197782 T^{4} - 289873265881071449 T^{6} + 643454197782 p^{6} T^{8} - 925629 p^{12} T^{10} + p^{18} T^{12} \)
79 \( 1 - 806709 T^{2} + 543912922362 T^{4} - 377274863700554573 T^{6} + 543912922362 p^{6} T^{8} - 806709 p^{12} T^{10} + p^{18} T^{12} \)
83 \( ( 1 + 12 T + 858849 T^{2} - 294490104 T^{3} + 858849 p^{3} T^{4} + 12 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
89 \( 1 - 2840157 T^{2} + 3962760458166 T^{4} - 3458858839547245369 T^{6} + 3962760458166 p^{6} T^{8} - 2840157 p^{12} T^{10} + p^{18} T^{12} \)
97 \( 1 - 4437342 T^{2} + 9010629855423 T^{4} - 10556356459893838148 T^{6} + 9010629855423 p^{6} T^{8} - 4437342 p^{12} T^{10} + p^{18} T^{12} \)
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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

−5.20371693911439541437277001576, −4.84647053040762145181068244866, −4.75018006116760922871491012048, −4.54770489748745336435564812274, −4.52565927415771457769916380108, −4.27175507733154495878521667193, −4.23519966769488392785329692927, −3.96596336055300683599093246538, −3.61186601940660291716316729121, −3.43547386268966083513596547427, −3.26697315709471011436554128730, −3.20994138113759557299512675351, −3.08496325542176949473423456970, −2.49788035714214858410299019745, −2.49496298323615690000981164823, −2.42926168752512248159070437664, −2.01287703836785589823508837111, −1.69222448816147821675605662581, −1.61772421812861114649490361161, −1.15367858243237165787230583014, −0.949824214590563073319197796965, −0.75237114078050223996833044061, −0.70327154519702883421232266857, −0.56960358216207807276808639931, −0.31101297883909133383863571838, 0.31101297883909133383863571838, 0.56960358216207807276808639931, 0.70327154519702883421232266857, 0.75237114078050223996833044061, 0.949824214590563073319197796965, 1.15367858243237165787230583014, 1.61772421812861114649490361161, 1.69222448816147821675605662581, 2.01287703836785589823508837111, 2.42926168752512248159070437664, 2.49496298323615690000981164823, 2.49788035714214858410299019745, 3.08496325542176949473423456970, 3.20994138113759557299512675351, 3.26697315709471011436554128730, 3.43547386268966083513596547427, 3.61186601940660291716316729121, 3.96596336055300683599093246538, 4.23519966769488392785329692927, 4.27175507733154495878521667193, 4.52565927415771457769916380108, 4.54770489748745336435564812274, 4.75018006116760922871491012048, 4.84647053040762145181068244866, 5.20371693911439541437277001576

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

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