Properties

Label 12-8470e6-1.1-c1e6-0-0
Degree $12$
Conductor $3.692\times 10^{23}$
Sign $1$
Analytic cond. $9.57112\times 10^{10}$
Root an. cond. $8.22394$
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  − 6·2-s − 5·3-s + 21·4-s − 6·5-s + 30·6-s − 6·7-s − 56·8-s + 9·9-s + 36·10-s − 105·12-s + 36·14-s + 30·15-s + 126·16-s + 15·17-s − 54·18-s + 13·19-s − 126·20-s + 30·21-s − 2·23-s + 280·24-s + 21·25-s − 7·27-s − 126·28-s + 4·29-s − 180·30-s − 2·31-s − 252·32-s + ⋯
L(s)  = 1  − 4.24·2-s − 2.88·3-s + 21/2·4-s − 2.68·5-s + 12.2·6-s − 2.26·7-s − 19.7·8-s + 3·9-s + 11.3·10-s − 30.3·12-s + 9.62·14-s + 7.74·15-s + 63/2·16-s + 3.63·17-s − 12.7·18-s + 2.98·19-s − 28.1·20-s + 6.54·21-s − 0.417·23-s + 57.1·24-s + 21/5·25-s − 1.34·27-s − 23.8·28-s + 0.742·29-s − 32.8·30-s − 0.359·31-s − 44.5·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(9.57112\times 10^{10}\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.06777511059\)
\(L(\frac12)\) \(\approx\) \(0.06777511059\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
5 \( ( 1 + T )^{6} \)
7 \( ( 1 + T )^{6} \)
11 \( 1 \)
good3 \( 1 + 5 T + 16 T^{2} + 14 p T^{3} + 97 T^{4} + 196 T^{5} + 359 T^{6} + 196 p T^{7} + 97 p^{2} T^{8} + 14 p^{4} T^{9} + 16 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 40 T^{2} + 34 T^{3} + 875 T^{4} + 1070 T^{5} + 13068 T^{6} + 1070 p T^{7} + 875 p^{2} T^{8} + 34 p^{3} T^{9} + 40 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 15 T + 138 T^{2} - 970 T^{3} + 5707 T^{4} - 28520 T^{5} + 124645 T^{6} - 28520 p T^{7} + 5707 p^{2} T^{8} - 970 p^{3} T^{9} + 138 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 13 T + 154 T^{2} - 1132 T^{3} + 7763 T^{4} - 40096 T^{5} + 197123 T^{6} - 40096 p T^{7} + 7763 p^{2} T^{8} - 1132 p^{3} T^{9} + 154 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 2 T + 58 T^{2} + 254 T^{3} + 2239 T^{4} + 7596 T^{5} + 69484 T^{6} + 7596 p T^{7} + 2239 p^{2} T^{8} + 254 p^{3} T^{9} + 58 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 4 T + 122 T^{2} - 510 T^{3} + 7151 T^{4} - 27354 T^{5} + 257848 T^{6} - 27354 p T^{7} + 7151 p^{2} T^{8} - 510 p^{3} T^{9} + 122 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 2 T + 70 T^{2} + 48 T^{3} + 2507 T^{4} + 2554 T^{5} + 82896 T^{6} + 2554 p T^{7} + 2507 p^{2} T^{8} + 48 p^{3} T^{9} + 70 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 160 T^{2} - 22 T^{3} + 12535 T^{4} - 2030 T^{5} + 583972 T^{6} - 2030 p T^{7} + 12535 p^{2} T^{8} - 22 p^{3} T^{9} + 160 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 17 T + 292 T^{2} - 3128 T^{3} + 32117 T^{4} - 245654 T^{5} + 1786223 T^{6} - 245654 p T^{7} + 32117 p^{2} T^{8} - 3128 p^{3} T^{9} + 292 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 15 T + 294 T^{2} + 3028 T^{3} + 33531 T^{4} + 252386 T^{5} + 1955439 T^{6} + 252386 p T^{7} + 33531 p^{2} T^{8} + 3028 p^{3} T^{9} + 294 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 18 T + 6 p T^{2} + 3342 T^{3} + 33743 T^{4} + 282684 T^{5} + 2116396 T^{6} + 282684 p T^{7} + 33743 p^{2} T^{8} + 3342 p^{3} T^{9} + 6 p^{5} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 22 T + 326 T^{2} - 3856 T^{3} + 36371 T^{4} - 301158 T^{5} + 2323416 T^{6} - 301158 p T^{7} + 36371 p^{2} T^{8} - 3856 p^{3} T^{9} + 326 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 7 T + 134 T^{2} + 868 T^{3} + 11523 T^{4} + 64404 T^{5} + 783443 T^{6} + 64404 p T^{7} + 11523 p^{2} T^{8} + 868 p^{3} T^{9} + 134 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 20 T + 326 T^{2} - 2938 T^{3} + 23295 T^{4} - 112874 T^{5} + 847776 T^{6} - 112874 p T^{7} + 23295 p^{2} T^{8} - 2938 p^{3} T^{9} + 326 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 29 T + 572 T^{2} + 8250 T^{3} + 98265 T^{4} + 979044 T^{5} + 8617941 T^{6} + 979044 p T^{7} + 98265 p^{2} T^{8} + 8250 p^{3} T^{9} + 572 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 2 T + 258 T^{2} + 560 T^{3} + 31551 T^{4} + 66622 T^{5} + 2594952 T^{6} + 66622 p T^{7} + 31551 p^{2} T^{8} + 560 p^{3} T^{9} + 258 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + T + 172 T^{2} + 576 T^{3} + 22731 T^{4} + 59064 T^{5} + 1971113 T^{6} + 59064 p T^{7} + 22731 p^{2} T^{8} + 576 p^{3} T^{9} + 172 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 12 T + 390 T^{2} - 3858 T^{3} + 70439 T^{4} - 554166 T^{5} + 7197976 T^{6} - 554166 p T^{7} + 70439 p^{2} T^{8} - 3858 p^{3} T^{9} + 390 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 35 T + 846 T^{2} - 14848 T^{3} + 212607 T^{4} - 2493634 T^{5} + 24781859 T^{6} - 2493634 p T^{7} + 212607 p^{2} T^{8} - 14848 p^{3} T^{9} + 846 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 29 T + 654 T^{2} + 9894 T^{3} + 136223 T^{4} + 1508924 T^{5} + 15667235 T^{6} + 1508924 p T^{7} + 136223 p^{2} T^{8} + 9894 p^{3} T^{9} + 654 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 19 T + 262 T^{2} + 2172 T^{3} + 28259 T^{4} + 364562 T^{5} + 4760381 T^{6} + 364562 p T^{7} + 28259 p^{2} T^{8} + 2172 p^{3} T^{9} + 262 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
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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

−3.85912651064508904215666982213, −3.60620120954987100826379825352, −3.58001718870563191857386375534, −3.42300869797423318968291142603, −3.34252600544301605356048534136, −3.31671102426663515532831403733, −3.19672527139322861718984864475, −2.92312093907896184443038530007, −2.73095888432387539945555709222, −2.64977687368417988506175998044, −2.64704577692784760879671168494, −2.44839138165857899433163257680, −2.43078767154967077120453590913, −1.75016932311114465513670479677, −1.66767406891857381365698416497, −1.64187935428550722550694889801, −1.38835363507498415084862762288, −1.20790001071356761489827318419, −1.19840060138804024281543332027, −0.842735187868477718277935452688, −0.67944815951468018837689731590, −0.58551640801742939130316041430, −0.49859037467705444041942821985, −0.34332369229951145774238037034, −0.20629108514192857871444970373, 0.20629108514192857871444970373, 0.34332369229951145774238037034, 0.49859037467705444041942821985, 0.58551640801742939130316041430, 0.67944815951468018837689731590, 0.842735187868477718277935452688, 1.19840060138804024281543332027, 1.20790001071356761489827318419, 1.38835363507498415084862762288, 1.64187935428550722550694889801, 1.66767406891857381365698416497, 1.75016932311114465513670479677, 2.43078767154967077120453590913, 2.44839138165857899433163257680, 2.64704577692784760879671168494, 2.64977687368417988506175998044, 2.73095888432387539945555709222, 2.92312093907896184443038530007, 3.19672527139322861718984864475, 3.31671102426663515532831403733, 3.34252600544301605356048534136, 3.42300869797423318968291142603, 3.58001718870563191857386375534, 3.60620120954987100826379825352, 3.85912651064508904215666982213

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

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