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Properties

Label 2-12-12.11-c9-0-0
Degree $2$
Conductor $12$
Sign $-0.430 - 0.902i$
Analytic cond. $6.18043$
Root an. cond. $2.48604$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.73 − 20.8i)2-s + (−132. − 45.7i)3-s + (−359. − 364. i)4-s + 1.83e3i·5-s + (−2.11e3 + 2.36e3i)6-s − 1.30e3i·7-s + (−1.07e4 + 4.31e3i)8-s + (1.54e4 + 1.21e4i)9-s + (3.82e4 + 1.60e4i)10-s − 6.70e4·11-s + (3.09e4 + 6.48e4i)12-s − 1.28e5·13-s + (−2.72e4 − 1.13e4i)14-s + (8.38e4 − 2.42e5i)15-s + (−3.89e3 + 2.62e5i)16-s − 7.87e4i·17-s + ⋯
L(s)  = 1  + (0.386 − 0.922i)2-s + (−0.945 − 0.326i)3-s + (−0.701 − 0.712i)4-s + 1.31i·5-s + (−0.666 + 0.745i)6-s − 0.205i·7-s + (−0.928 + 0.372i)8-s + (0.787 + 0.616i)9-s + (1.20 + 0.506i)10-s − 1.38·11-s + (0.430 + 0.902i)12-s − 1.24·13-s + (−0.189 − 0.0792i)14-s + (0.427 − 1.23i)15-s + (−0.0148 + 0.999i)16-s − 0.228i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.430 - 0.902i$
Analytic conductor: \(6.18043\)
Root analytic conductor: \(2.48604\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :9/2),\ -0.430 - 0.902i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0186661 + 0.0295995i\)
\(L(\frac12)\) \(\approx\) \(0.0186661 + 0.0295995i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8.73 + 20.8i)T \)
3 \( 1 + (132. + 45.7i)T \)
good5 \( 1 - 1.83e3iT - 1.95e6T^{2} \)
7 \( 1 + 1.30e3iT - 4.03e7T^{2} \)
11 \( 1 + 6.70e4T + 2.35e9T^{2} \)
13 \( 1 + 1.28e5T + 1.06e10T^{2} \)
17 \( 1 + 7.87e4iT - 1.18e11T^{2} \)
19 \( 1 + 7.05e5iT - 3.22e11T^{2} \)
23 \( 1 + 3.02e5T + 1.80e12T^{2} \)
29 \( 1 - 1.71e5iT - 1.45e13T^{2} \)
31 \( 1 - 8.57e6iT - 2.64e13T^{2} \)
37 \( 1 + 2.65e6T + 1.29e14T^{2} \)
41 \( 1 + 3.10e7iT - 3.27e14T^{2} \)
43 \( 1 - 5.87e6iT - 5.02e14T^{2} \)
47 \( 1 + 3.06e7T + 1.11e15T^{2} \)
53 \( 1 - 6.60e4iT - 3.29e15T^{2} \)
59 \( 1 + 3.52e7T + 8.66e15T^{2} \)
61 \( 1 + 8.17e7T + 1.16e16T^{2} \)
67 \( 1 - 4.25e7iT - 2.72e16T^{2} \)
71 \( 1 - 7.59e7T + 4.58e16T^{2} \)
73 \( 1 - 1.16e8T + 5.88e16T^{2} \)
79 \( 1 - 4.69e8iT - 1.19e17T^{2} \)
83 \( 1 + 4.67e8T + 1.86e17T^{2} \)
89 \( 1 + 3.29e7iT - 3.50e17T^{2} \)
97 \( 1 + 1.39e9T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.46913194921818546170096367666, −17.63402006091623600715214894411, −15.44417967111982508063280351794, −13.88428087590399919868586128616, −12.46024355188587509315630520224, −11.02105796889618295984740093776, −10.20991522269422303272615026147, −7.06667932091827278281602513179, −5.11583571239722753380757484901, −2.59247573877920548520744553019, 0.01890894060798263294235879070, 4.63049659230756457058207221853, 5.68968362101108100337335496991, 7.901147882215655675881383070565, 9.751888036234792420120948935701, 12.15851246096818182634933756097, 13.03484989842327201674812142778, 15.13716256260925601929756506146, 16.34073940455609231169420884980, 17.04542594059907990623490585805

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