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Properties

Label 12-7616e6-1.1-c1e6-0-8
Degree $12$
Conductor $1.951\times 10^{23}$
Sign $1$
Analytic cond. $5.05852\times 10^{10}$
Root an. cond. $7.79833$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 6·5-s − 6·7-s − 5·9-s − 2·11-s + 4·13-s − 12·15-s − 6·17-s − 10·19-s + 12·21-s − 4·23-s + 5·25-s + 12·27-s + 14·29-s − 8·31-s + 4·33-s − 36·35-s + 4·37-s − 8·39-s − 2·41-s − 8·43-s − 30·45-s + 2·47-s + 21·49-s + 12·51-s + 10·53-s − 12·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.68·5-s − 2.26·7-s − 5/3·9-s − 0.603·11-s + 1.10·13-s − 3.09·15-s − 1.45·17-s − 2.29·19-s + 2.61·21-s − 0.834·23-s + 25-s + 2.30·27-s + 2.59·29-s − 1.43·31-s + 0.696·33-s − 6.08·35-s + 0.657·37-s − 1.28·39-s − 0.312·41-s − 1.21·43-s − 4.47·45-s + 0.291·47-s + 3·49-s + 1.68·51-s + 1.37·53-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{36} \cdot 7^{6} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(5.05852\times 10^{10}\)
Root analytic conductor: \(7.79833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{36} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 + T )^{6} \)
17 \( ( 1 + T )^{6} \)
good3 \( 1 + 2 T + p^{2} T^{2} + 16 T^{3} + 16 p T^{4} + 68 T^{5} + 56 p T^{6} + 68 p T^{7} + 16 p^{3} T^{8} + 16 p^{3} T^{9} + p^{6} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - 6 T + 31 T^{2} - 122 T^{3} + 16 p^{2} T^{4} - 1106 T^{5} + 2688 T^{6} - 1106 p T^{7} + 16 p^{4} T^{8} - 122 p^{3} T^{9} + 31 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 2 T + 32 T^{2} + 18 T^{3} + 515 T^{4} - 40 T^{5} + 6232 T^{6} - 40 p T^{7} + 515 p^{2} T^{8} + 18 p^{3} T^{9} + 32 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 4 T + 4 p T^{2} - 116 T^{3} + 971 T^{4} - 80 p T^{5} + 12048 T^{6} - 80 p^{2} T^{7} + 971 p^{2} T^{8} - 116 p^{3} T^{9} + 4 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 10 T + 82 T^{2} + 326 T^{3} + 1027 T^{4} - 852 T^{5} - 6444 T^{6} - 852 p T^{7} + 1027 p^{2} T^{8} + 326 p^{3} T^{9} + 82 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 4 T + 2 p T^{2} + 148 T^{3} + 1163 T^{4} + 2656 T^{5} + 23980 T^{6} + 2656 p T^{7} + 1163 p^{2} T^{8} + 148 p^{3} T^{9} + 2 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 14 T + 220 T^{2} - 1906 T^{3} + 17179 T^{4} - 106472 T^{5} + 675888 T^{6} - 106472 p T^{7} + 17179 p^{2} T^{8} - 1906 p^{3} T^{9} + 220 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 89 T^{2} + 664 T^{3} + 4656 T^{4} + 25704 T^{5} + 169068 T^{6} + 25704 p T^{7} + 4656 p^{2} T^{8} + 664 p^{3} T^{9} + 89 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 4 T + 122 T^{2} - 500 T^{3} + 8227 T^{4} - 28312 T^{5} + 365892 T^{6} - 28312 p T^{7} + 8227 p^{2} T^{8} - 500 p^{3} T^{9} + 122 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 2 T + 195 T^{2} + 182 T^{3} + 16804 T^{4} + 6552 T^{5} + 861264 T^{6} + 6552 p T^{7} + 16804 p^{2} T^{8} + 182 p^{3} T^{9} + 195 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 8 T + 187 T^{2} + 1114 T^{3} + 16156 T^{4} + 78342 T^{5} + 862296 T^{6} + 78342 p T^{7} + 16156 p^{2} T^{8} + 1114 p^{3} T^{9} + 187 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 2 T + 192 T^{2} - 250 T^{3} + 17923 T^{4} - 17448 T^{5} + 1041080 T^{6} - 17448 p T^{7} + 17923 p^{2} T^{8} - 250 p^{3} T^{9} + 192 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 10 T + 225 T^{2} - 1956 T^{3} + 23880 T^{4} - 174814 T^{5} + 1566748 T^{6} - 174814 p T^{7} + 23880 p^{2} T^{8} - 1956 p^{3} T^{9} + 225 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 14 T + 214 T^{2} + 2370 T^{3} + 25991 T^{4} + 223436 T^{5} + 1918964 T^{6} + 223436 p T^{7} + 25991 p^{2} T^{8} + 2370 p^{3} T^{9} + 214 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 20 T + 263 T^{2} - 1898 T^{3} + 3984 T^{4} + 82348 T^{5} - 1046808 T^{6} + 82348 p T^{7} + 3984 p^{2} T^{8} - 1898 p^{3} T^{9} + 263 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T + 271 T^{2} + 1610 T^{3} + 35028 T^{4} + 188628 T^{5} + 2861664 T^{6} + 188628 p T^{7} + 35028 p^{2} T^{8} + 1610 p^{3} T^{9} + 271 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 20 T + 446 T^{2} + 5884 T^{3} + 79003 T^{4} + 765512 T^{5} + 7422124 T^{6} + 765512 p T^{7} + 79003 p^{2} T^{8} + 5884 p^{3} T^{9} + 446 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 6 T + 347 T^{2} + 1634 T^{3} + 53720 T^{4} + 200976 T^{5} + 4919912 T^{6} + 200976 p T^{7} + 53720 p^{2} T^{8} + 1634 p^{3} T^{9} + 347 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 38 T + 740 T^{2} + 9566 T^{3} + 97155 T^{4} + 876392 T^{5} + 7764912 T^{6} + 876392 p T^{7} + 97155 p^{2} T^{8} + 9566 p^{3} T^{9} + 740 p^{4} T^{10} + 38 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 16 T + 376 T^{2} + 5088 T^{3} + 66491 T^{4} + 748248 T^{5} + 7035736 T^{6} + 748248 p T^{7} + 66491 p^{2} T^{8} + 5088 p^{3} T^{9} + 376 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 2 T + 196 T^{2} - 1210 T^{3} + 25163 T^{4} - 107880 T^{5} + 3544352 T^{6} - 107880 p T^{7} + 25163 p^{2} T^{8} - 1210 p^{3} T^{9} + 196 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 12 T + 435 T^{2} + 4682 T^{3} + 89128 T^{4} + 812778 T^{5} + 10912024 T^{6} + 812778 p T^{7} + 89128 p^{2} T^{8} + 4682 p^{3} T^{9} + 435 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.39517196128295950093334351060, −4.21634493487487259845050060358, −4.02600795234371791959793060936, −3.88682167798054317988311265563, −3.88391083015159021515031194887, −3.85922343394869849497238361758, −3.80193455783976329715711282519, −3.47108534005500514051923079877, −3.12119893011401368911991680894, −3.01952069181727032735100116603, −2.95029509755443634902852653952, −2.93882944246452114614097312852, −2.80889152618158415774380077422, −2.53861546274334640541523157718, −2.42903550066146997906293540279, −2.35530048502618645618380651509, −2.29737202665912778932448437474, −2.11036794512578035012628245033, −1.93436663206746478905283041221, −1.68255661731242692862343202181, −1.60035557518999633928839548955, −1.29594830470763483605472208212, −1.16350973866516028518207345919, −1.07536869742938125783854784678, −0.955809092500490405412692096853, 0, 0, 0, 0, 0, 0, 0.955809092500490405412692096853, 1.07536869742938125783854784678, 1.16350973866516028518207345919, 1.29594830470763483605472208212, 1.60035557518999633928839548955, 1.68255661731242692862343202181, 1.93436663206746478905283041221, 2.11036794512578035012628245033, 2.29737202665912778932448437474, 2.35530048502618645618380651509, 2.42903550066146997906293540279, 2.53861546274334640541523157718, 2.80889152618158415774380077422, 2.93882944246452114614097312852, 2.95029509755443634902852653952, 3.01952069181727032735100116603, 3.12119893011401368911991680894, 3.47108534005500514051923079877, 3.80193455783976329715711282519, 3.85922343394869849497238361758, 3.88391083015159021515031194887, 3.88682167798054317988311265563, 4.02600795234371791959793060936, 4.21634493487487259845050060358, 4.39517196128295950093334351060

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

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