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Properties

Label 12-3344e6-1.1-c1e6-0-4
Degree $12$
Conductor $1.398\times 10^{21}$
Sign $1$
Analytic cond. $3.62460\times 10^{8}$
Root an. cond. $5.16739$
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  − 4·3-s + 5-s + 9-s − 6·11-s − 4·15-s + 3·17-s − 6·19-s − 7·23-s − 13·25-s + 17·27-s − 4·29-s − 7·31-s + 24·33-s − 2·37-s + 7·41-s − 21·43-s + 45-s − 16·47-s − 20·49-s − 12·51-s + 17·53-s − 6·55-s + 24·57-s − 19·59-s − 6·61-s − 14·67-s + 28·69-s + ⋯
L(s)  = 1  − 2.30·3-s + 0.447·5-s + 1/3·9-s − 1.80·11-s − 1.03·15-s + 0.727·17-s − 1.37·19-s − 1.45·23-s − 2.59·25-s + 3.27·27-s − 0.742·29-s − 1.25·31-s + 4.17·33-s − 0.328·37-s + 1.09·41-s − 3.20·43-s + 0.149·45-s − 2.33·47-s − 2.85·49-s − 1.68·51-s + 2.33·53-s − 0.809·55-s + 3.17·57-s − 2.47·59-s − 0.768·61-s − 1.71·67-s + 3.37·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 11^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(3.62460\times 10^{8}\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{24} \cdot 11^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( 1 \)
11 \( ( 1 + T )^{6} \)
19 \( ( 1 + T )^{6} \)
good3 \( 1 + 4 T + 5 p T^{2} + 13 p T^{3} + 95 T^{4} + 194 T^{5} + 362 T^{6} + 194 p T^{7} + 95 p^{2} T^{8} + 13 p^{4} T^{9} + 5 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - T + 14 T^{2} - 2 p T^{3} + 92 T^{4} - 29 T^{5} + 466 T^{6} - 29 p T^{7} + 92 p^{2} T^{8} - 2 p^{4} T^{9} + 14 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 20 T^{2} - 3 T^{3} + 25 p T^{4} - 57 T^{5} + 1140 T^{6} - 57 p T^{7} + 25 p^{3} T^{8} - 3 p^{3} T^{9} + 20 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 56 T^{2} + 3 T^{3} + 1447 T^{4} + 111 T^{5} + 23052 T^{6} + 111 p T^{7} + 1447 p^{2} T^{8} + 3 p^{3} T^{9} + 56 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 3 T + 86 T^{2} - 233 T^{3} + 3279 T^{4} - 7564 T^{5} + 71588 T^{6} - 7564 p T^{7} + 3279 p^{2} T^{8} - 233 p^{3} T^{9} + 86 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T + 5 p T^{2} + 654 T^{3} + 5997 T^{4} + 26983 T^{5} + 178654 T^{6} + 26983 p T^{7} + 5997 p^{2} T^{8} + 654 p^{3} T^{9} + 5 p^{5} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 4 T + 56 T^{2} + 67 T^{3} + 183 T^{4} - 3677 T^{5} - 27356 T^{6} - 3677 p T^{7} + 183 p^{2} T^{8} + 67 p^{3} T^{9} + 56 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 7 T + 108 T^{2} + 664 T^{3} + 5928 T^{4} + 34277 T^{5} + 225226 T^{6} + 34277 p T^{7} + 5928 p^{2} T^{8} + 664 p^{3} T^{9} + 108 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 2 T + 187 T^{2} + 376 T^{3} + 15563 T^{4} + 27918 T^{5} + 740834 T^{6} + 27918 p T^{7} + 15563 p^{2} T^{8} + 376 p^{3} T^{9} + 187 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 7 T + 225 T^{2} - 1251 T^{3} + 21703 T^{4} - 95550 T^{5} + 1160782 T^{6} - 95550 p T^{7} + 21703 p^{2} T^{8} - 1251 p^{3} T^{9} + 225 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 21 T + 287 T^{2} + 2191 T^{3} + 10663 T^{4} + 10046 T^{5} - 84622 T^{6} + 10046 p T^{7} + 10663 p^{2} T^{8} + 2191 p^{3} T^{9} + 287 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 16 T + 290 T^{2} + 3088 T^{3} + 33823 T^{4} + 265728 T^{5} + 2101692 T^{6} + 265728 p T^{7} + 33823 p^{2} T^{8} + 3088 p^{3} T^{9} + 290 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 17 T + 344 T^{2} - 3901 T^{3} + 46719 T^{4} - 388982 T^{5} + 3313792 T^{6} - 388982 p T^{7} + 46719 p^{2} T^{8} - 3901 p^{3} T^{9} + 344 p^{4} T^{10} - 17 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 19 T + 259 T^{2} + 3008 T^{3} + 32403 T^{4} + 289897 T^{5} + 2374706 T^{6} + 289897 p T^{7} + 32403 p^{2} T^{8} + 3008 p^{3} T^{9} + 259 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T + 4 p T^{2} + 22 p T^{3} + 29111 T^{4} + 138948 T^{5} + 2171864 T^{6} + 138948 p T^{7} + 29111 p^{2} T^{8} + 22 p^{4} T^{9} + 4 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 14 T + 235 T^{2} + 2073 T^{3} + 23803 T^{4} + 172820 T^{5} + 1786642 T^{6} + 172820 p T^{7} + 23803 p^{2} T^{8} + 2073 p^{3} T^{9} + 235 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + T + 222 T^{2} + 1004 T^{3} + 26366 T^{4} + 135647 T^{5} + 2298562 T^{6} + 135647 p T^{7} + 26366 p^{2} T^{8} + 1004 p^{3} T^{9} + 222 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 5 T + 266 T^{2} + 1737 T^{3} + 34999 T^{4} + 238442 T^{5} + 3055996 T^{6} + 238442 p T^{7} + 34999 p^{2} T^{8} + 1737 p^{3} T^{9} + 266 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 308 T^{2} + 444 T^{3} + 47999 T^{4} + 68652 T^{5} + 4774360 T^{6} + 68652 p T^{7} + 47999 p^{2} T^{8} + 444 p^{3} T^{9} + 308 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 11 T + 259 T^{2} + 2809 T^{3} + 45125 T^{4} + 383834 T^{5} + 4460318 T^{6} + 383834 p T^{7} + 45125 p^{2} T^{8} + 2809 p^{3} T^{9} + 259 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 12 T + 319 T^{2} - 3584 T^{3} + 55795 T^{4} - 517308 T^{5} + 6160106 T^{6} - 517308 p T^{7} + 55795 p^{2} T^{8} - 3584 p^{3} T^{9} + 319 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 14 T + 335 T^{2} - 1708 T^{3} + 21123 T^{4} + 187618 T^{5} - 368278 T^{6} + 187618 p T^{7} + 21123 p^{2} T^{8} - 1708 p^{3} T^{9} + 335 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.01172175451105415746864038396, −4.89738939191696392152618624346, −4.57235371442327165379009443948, −4.54602604993324896542166707725, −4.41398998672318018402272843604, −4.29008206148895077563153750836, −4.12816417671618833858744102349, −3.73743409446342192080737076317, −3.57566381483189874665066181538, −3.53371235132391945990980449805, −3.52845650081143516568760419031, −3.36437545393694992129232941532, −3.30086376738166310562073301557, −2.77582724676706233568486880840, −2.77366239566713966114464564082, −2.67284895567722011160111067536, −2.45967403630915437610332378810, −2.17147555669117040830742153596, −2.15586229890968112127383941388, −1.98385446389458977507213489795, −1.66367824486932134097459394605, −1.64989814659290967461581568727, −1.36049172356529115273911667401, −1.10117988099581115066112756228, −1.06075971026539627873777655255, 0, 0, 0, 0, 0, 0, 1.06075971026539627873777655255, 1.10117988099581115066112756228, 1.36049172356529115273911667401, 1.64989814659290967461581568727, 1.66367824486932134097459394605, 1.98385446389458977507213489795, 2.15586229890968112127383941388, 2.17147555669117040830742153596, 2.45967403630915437610332378810, 2.67284895567722011160111067536, 2.77366239566713966114464564082, 2.77582724676706233568486880840, 3.30086376738166310562073301557, 3.36437545393694992129232941532, 3.52845650081143516568760419031, 3.53371235132391945990980449805, 3.57566381483189874665066181538, 3.73743409446342192080737076317, 4.12816417671618833858744102349, 4.29008206148895077563153750836, 4.41398998672318018402272843604, 4.54602604993324896542166707725, 4.57235371442327165379009443948, 4.89738939191696392152618624346, 5.01172175451105415746864038396

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

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