Properties

Label 12-8470e6-1.1-c1e6-0-8
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 $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 3-s + 21·4-s + 6·5-s − 6·6-s − 6·7-s − 56·8-s − 9-s − 36·10-s + 21·12-s − 2·13-s + 36·14-s + 6·15-s + 126·16-s − 7·17-s + 6·18-s − 11·19-s + 126·20-s − 6·21-s − 6·23-s − 56·24-s + 21·25-s + 12·26-s − 27-s − 126·28-s − 2·29-s − 36·30-s + ⋯
L(s)  = 1  − 4.24·2-s + 0.577·3-s + 21/2·4-s + 2.68·5-s − 2.44·6-s − 2.26·7-s − 19.7·8-s − 1/3·9-s − 11.3·10-s + 6.06·12-s − 0.554·13-s + 9.62·14-s + 1.54·15-s + 63/2·16-s − 1.69·17-s + 1.41·18-s − 2.52·19-s + 28.1·20-s − 1.30·21-s − 1.25·23-s − 11.4·24-s + 21/5·25-s + 2.35·26-s − 0.192·27-s − 23.8·28-s − 0.371·29-s − 6.57·30-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: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( ( 1 + T )^{6} \)
5 \( ( 1 - T )^{6} \)
7 \( ( 1 + T )^{6} \)
11 \( 1 \)
good3 \( 1 - T + 2 T^{2} - 2 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + 43 T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} - 2 p^{3} T^{9} + 2 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 2 T + 46 T^{2} + 36 T^{3} + 1031 T^{4} + 418 T^{5} + 16136 T^{6} + 418 p T^{7} + 1031 p^{2} T^{8} + 36 p^{3} T^{9} + 46 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 7 T + 82 T^{2} + 412 T^{3} + 2667 T^{4} + 10758 T^{5} + 53119 T^{6} + 10758 p T^{7} + 2667 p^{2} T^{8} + 412 p^{3} T^{9} + 82 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 11 T + 134 T^{2} + 868 T^{3} + 6039 T^{4} + 28176 T^{5} + 145727 T^{6} + 28176 p T^{7} + 6039 p^{2} T^{8} + 868 p^{3} T^{9} + 134 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 82 T^{2} + 22 p T^{3} + 3551 T^{4} + 19524 T^{5} + 99068 T^{6} + 19524 p T^{7} + 3551 p^{2} T^{8} + 22 p^{4} T^{9} + 82 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 2 T + 112 T^{2} + 112 T^{3} + 6199 T^{4} + 4674 T^{5} + 222956 T^{6} + 4674 p T^{7} + 6199 p^{2} T^{8} + 112 p^{3} T^{9} + 112 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 144 T^{2} + 58 T^{3} + 9523 T^{4} + 4926 T^{5} + 372956 T^{6} + 4926 p T^{7} + 9523 p^{2} T^{8} + 58 p^{3} T^{9} + 144 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 + 14 T + 170 T^{2} + 1256 T^{3} + 8519 T^{4} + 43862 T^{5} + 266248 T^{6} + 43862 p T^{7} + 8519 p^{2} T^{8} + 1256 p^{3} T^{9} + 170 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 13 T + 262 T^{2} + 2314 T^{3} + 26585 T^{4} + 175582 T^{5} + 857 p^{2} T^{6} + 175582 p T^{7} + 26585 p^{2} T^{8} + 2314 p^{3} T^{9} + 262 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 19 T + 278 T^{2} + 2670 T^{3} + 21395 T^{4} + 144524 T^{5} + 950221 T^{6} + 144524 p T^{7} + 21395 p^{2} T^{8} + 2670 p^{3} T^{9} + 278 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 22 T + 322 T^{2} - 3242 T^{3} + 27887 T^{4} - 204308 T^{5} + 1456284 T^{6} - 204308 p T^{7} + 27887 p^{2} T^{8} - 3242 p^{3} T^{9} + 322 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 10 T + 268 T^{2} + 2312 T^{3} + 32335 T^{4} + 229038 T^{5} + 2219396 T^{6} + 229038 p T^{7} + 32335 p^{2} T^{8} + 2312 p^{3} T^{9} + 268 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 7 T + 282 T^{2} + 1772 T^{3} + 35099 T^{4} + 191984 T^{5} + 2589251 T^{6} + 191984 p T^{7} + 35099 p^{2} T^{8} + 1772 p^{3} T^{9} + 282 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 22 T + 410 T^{2} + 4232 T^{3} + 41271 T^{4} + 272662 T^{5} + 2325648 T^{6} + 272662 p T^{7} + 41271 p^{2} T^{8} + 4232 p^{3} T^{9} + 410 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 5 T + 176 T^{2} - 496 T^{3} + 12781 T^{4} + 15162 T^{5} + 728231 T^{6} + 15162 p T^{7} + 12781 p^{2} T^{8} - 496 p^{3} T^{9} + 176 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 8 T + 252 T^{2} - 1754 T^{3} + 29715 T^{4} - 177122 T^{5} + 2381652 T^{6} - 177122 p T^{7} + 29715 p^{2} T^{8} - 1754 p^{3} T^{9} + 252 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 13 T + 268 T^{2} + 2302 T^{3} + 28743 T^{4} + 204130 T^{5} + 2189947 T^{6} + 204130 p T^{7} + 28743 p^{2} T^{8} + 2302 p^{3} T^{9} + 268 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 16 T + 224 T^{2} - 2498 T^{3} + 34039 T^{4} - 291126 T^{5} + 2720252 T^{6} - 291126 p T^{7} + 34039 p^{2} T^{8} - 2498 p^{3} T^{9} + 224 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 5 T + 304 T^{2} - 2048 T^{3} + 45651 T^{4} - 335256 T^{5} + 4479499 T^{6} - 335256 p T^{7} + 45651 p^{2} T^{8} - 2048 p^{3} T^{9} + 304 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - T + 196 T^{2} - 624 T^{3} + 29603 T^{4} - 71168 T^{5} + 3211265 T^{6} - 71168 p T^{7} + 29603 p^{2} T^{8} - 624 p^{3} T^{9} + 196 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T + 342 T^{2} - 982 T^{3} + 39927 T^{4} - 437408 T^{5} + 3130779 T^{6} - 437408 p T^{7} + 39927 p^{2} T^{8} - 982 p^{3} T^{9} + 342 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.28452637516382305211332987534, −4.03237098851567463609847853469, −4.00284109873171763087394173816, −3.79598810142175860538031767559, −3.63926556877605486613028881474, −3.63363041504930315044552060629, −3.50848169945912280899604275540, −3.02697828856266508585647460515, −2.95984569850982958694848089400, −2.94328907284521327072761639341, −2.88299472200833626955013392851, −2.85963721586264389540227475931, −2.68765335910088442537706570873, −2.35575016827317682901191023821, −2.26420506093444842662014837519, −2.07357966365610978109884131407, −2.05071609056907941882022980827, −1.97652911839618267154678113838, −1.83893521082177798890848477504, −1.78245011799957503055671476228, −1.49261623687931561200443730167, −1.28151262377686490584864087089, −1.14365062572257572899502140903, −0.982979967444681421269924550978, −0.910397570636313685935187403424, 0, 0, 0, 0, 0, 0, 0.910397570636313685935187403424, 0.982979967444681421269924550978, 1.14365062572257572899502140903, 1.28151262377686490584864087089, 1.49261623687931561200443730167, 1.78245011799957503055671476228, 1.83893521082177798890848477504, 1.97652911839618267154678113838, 2.05071609056907941882022980827, 2.07357966365610978109884131407, 2.26420506093444842662014837519, 2.35575016827317682901191023821, 2.68765335910088442537706570873, 2.85963721586264389540227475931, 2.88299472200833626955013392851, 2.94328907284521327072761639341, 2.95984569850982958694848089400, 3.02697828856266508585647460515, 3.50848169945912280899604275540, 3.63363041504930315044552060629, 3.63926556877605486613028881474, 3.79598810142175860538031767559, 4.00284109873171763087394173816, 4.03237098851567463609847853469, 4.28452637516382305211332987534

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

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