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Properties

Label 12-5415e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.521\times 10^{22}$
Sign $1$
Analytic cond. $6.53511\times 10^{9}$
Root an. cond. $6.57563$
Motivic weight $1$
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  + 2-s − 6·3-s − 2·4-s − 6·5-s − 6·6-s + 4·7-s − 8-s + 21·9-s − 6·10-s − 14·11-s + 12·12-s + 8·13-s + 4·14-s + 36·15-s + 16-s + 3·17-s + 21·18-s + 12·20-s − 24·21-s − 14·22-s − 15·23-s + 6·24-s + 21·25-s + 8·26-s − 56·27-s − 8·28-s − 15·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 3.46·3-s − 4-s − 2.68·5-s − 2.44·6-s + 1.51·7-s − 0.353·8-s + 7·9-s − 1.89·10-s − 4.22·11-s + 3.46·12-s + 2.21·13-s + 1.06·14-s + 9.29·15-s + 1/4·16-s + 0.727·17-s + 4.94·18-s + 2.68·20-s − 5.23·21-s − 2.98·22-s − 3.12·23-s + 1.22·24-s + 21/5·25-s + 1.56·26-s − 10.7·27-s − 1.51·28-s − 2.78·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(6.53511\times 10^{9}\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8922273584\)
\(L(\frac12)\) \(\approx\) \(0.8922273584\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
5 \( ( 1 + T )^{6} \)
19 \( 1 \)
good2 \( 1 - T + 3 T^{2} - p^{2} T^{3} + p^{3} T^{4} - 7 T^{5} + 15 T^{6} - 7 p T^{7} + p^{5} T^{8} - p^{5} T^{9} + 3 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 4 T + 2 p T^{2} - 13 T^{3} + T^{4} + 239 T^{5} - 591 T^{6} + 239 p T^{7} + p^{2} T^{8} - 13 p^{3} T^{9} + 2 p^{5} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 14 T + 111 T^{2} + 629 T^{3} + 2853 T^{4} + 1008 p T^{5} + 38567 T^{6} + 1008 p^{2} T^{7} + 2853 p^{2} T^{8} + 629 p^{3} T^{9} + 111 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 63 T^{2} - 384 T^{3} + 1966 T^{4} - 8472 T^{5} + 33827 T^{6} - 8472 p T^{7} + 1966 p^{2} T^{8} - 384 p^{3} T^{9} + 63 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 3 T + 52 T^{2} - 128 T^{3} + 1372 T^{4} - 2367 T^{5} + 25829 T^{6} - 2367 p T^{7} + 1372 p^{2} T^{8} - 128 p^{3} T^{9} + 52 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 15 T + 170 T^{2} + 1376 T^{3} + 9420 T^{4} + 53885 T^{5} + 277993 T^{6} + 53885 p T^{7} + 9420 p^{2} T^{8} + 1376 p^{3} T^{9} + 170 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 15 T + 175 T^{2} + 1493 T^{3} + 11860 T^{4} + 75340 T^{5} + 442309 T^{6} + 75340 p T^{7} + 11860 p^{2} T^{8} + 1493 p^{3} T^{9} + 175 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 126 T^{2} + 236 T^{3} + 7714 T^{4} + 11843 T^{5} + 300653 T^{6} + 11843 p T^{7} + 7714 p^{2} T^{8} + 236 p^{3} T^{9} + 126 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 31 T + 528 T^{2} - 6541 T^{3} + 64309 T^{4} - 514156 T^{5} + 3414553 T^{6} - 514156 p T^{7} + 64309 p^{2} T^{8} - 6541 p^{3} T^{9} + 528 p^{4} T^{10} - 31 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 21 T + 358 T^{2} - 3889 T^{3} + 37579 T^{4} - 281812 T^{5} + 1993769 T^{6} - 281812 p T^{7} + 37579 p^{2} T^{8} - 3889 p^{3} T^{9} + 358 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 2 T + p T^{2} - 416 T^{3} + 1069 T^{4} - 6274 T^{5} + 189539 T^{6} - 6274 p T^{7} + 1069 p^{2} T^{8} - 416 p^{3} T^{9} + p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 22 T + 384 T^{2} + 4630 T^{3} + 48359 T^{4} + 406177 T^{5} + 3056729 T^{6} + 406177 p T^{7} + 48359 p^{2} T^{8} + 4630 p^{3} T^{9} + 384 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 12 T + 304 T^{2} - 52 p T^{3} + 38728 T^{4} - 273104 T^{5} + 2700449 T^{6} - 273104 p T^{7} + 38728 p^{2} T^{8} - 52 p^{4} T^{9} + 304 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 7 T + 301 T^{2} + 2097 T^{3} + 39912 T^{4} + 248992 T^{5} + 3024961 T^{6} + 248992 p T^{7} + 39912 p^{2} T^{8} + 2097 p^{3} T^{9} + 301 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 4 T + 161 T^{2} - 664 T^{3} + 278 p T^{4} - 60723 T^{5} + 1156827 T^{6} - 60723 p T^{7} + 278 p^{3} T^{8} - 664 p^{3} T^{9} + 161 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 9 T + 91 T^{2} - 417 T^{3} + 2118 T^{4} + 37478 T^{5} + 1044271 T^{6} + 37478 p T^{7} + 2118 p^{2} T^{8} - 417 p^{3} T^{9} + 91 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 9 T + 358 T^{2} + 2656 T^{3} + 58279 T^{4} + 4888 p T^{5} + 5356039 T^{6} + 4888 p^{2} T^{7} + 58279 p^{2} T^{8} + 2656 p^{3} T^{9} + 358 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T + 173 T^{2} + 690 T^{3} + 16220 T^{4} + 113887 T^{5} + 1210089 T^{6} + 113887 p T^{7} + 16220 p^{2} T^{8} + 690 p^{3} T^{9} + 173 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + T + 172 T^{2} - 321 T^{3} + 14973 T^{4} - 20078 T^{5} + 1315849 T^{6} - 20078 p T^{7} + 14973 p^{2} T^{8} - 321 p^{3} T^{9} + 172 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 27 T + 754 T^{2} + 12223 T^{3} + 189955 T^{4} + 2114092 T^{5} + 22287143 T^{6} + 2114092 p T^{7} + 189955 p^{2} T^{8} + 12223 p^{3} T^{9} + 754 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 37 T + 719 T^{2} - 6919 T^{3} + 7944 T^{4} + 798782 T^{5} - 11397233 T^{6} + 798782 p T^{7} + 7944 p^{2} T^{8} - 6919 p^{3} T^{9} + 719 p^{4} T^{10} - 37 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 23 T + 612 T^{2} - 8952 T^{3} + 138608 T^{4} - 1513579 T^{5} + 17268121 T^{6} - 1513579 p T^{7} + 138608 p^{2} T^{8} - 8952 p^{3} T^{9} + 612 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.23550141940230903249446991298, −3.99540614941716856671999565797, −3.97501844913141168759624628573, −3.94864874152709751026661817771, −3.94837236248480345138073698740, −3.64621678438022786473178524561, −3.62055659749591858853734802311, −3.19292178927679530735393581153, −3.08508503847585993649986233738, −3.01150243096176922524217997529, −3.00310857286439625422456043953, −2.42878133269364813577948992968, −2.38579409739198046733760992688, −2.35127293694830250300376874131, −2.22875199975904305615518648464, −1.72069822819598004833306838819, −1.62595944281385477164131169541, −1.59451187557406538282404576949, −1.49246518619878374763102094029, −1.07192868824478406956853280482, −0.802249690941055276210393104475, −0.52363031162643699217380839907, −0.42440025008301221476536822367, −0.39725981633737071657272328007, −0.36950775704898412008859467912, 0.36950775704898412008859467912, 0.39725981633737071657272328007, 0.42440025008301221476536822367, 0.52363031162643699217380839907, 0.802249690941055276210393104475, 1.07192868824478406956853280482, 1.49246518619878374763102094029, 1.59451187557406538282404576949, 1.62595944281385477164131169541, 1.72069822819598004833306838819, 2.22875199975904305615518648464, 2.35127293694830250300376874131, 2.38579409739198046733760992688, 2.42878133269364813577948992968, 3.00310857286439625422456043953, 3.01150243096176922524217997529, 3.08508503847585993649986233738, 3.19292178927679530735393581153, 3.62055659749591858853734802311, 3.64621678438022786473178524561, 3.94837236248480345138073698740, 3.94864874152709751026661817771, 3.97501844913141168759624628573, 3.99540614941716856671999565797, 4.23550141940230903249446991298

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

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