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Properties

Label 2-12-12.11-c9-0-7
Degree $2$
Conductor $12$
Sign $0.314 - 0.949i$
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  + (22.5 − 2.28i)2-s + (−69.9 + 121. i)3-s + (501. − 102. i)4-s + 852. i·5-s + (−1.29e3 + 2.89e3i)6-s + 9.64e3i·7-s + (1.10e4 − 3.46e3i)8-s + (−9.90e3 − 1.70e4i)9-s + (1.94e3 + 1.91e4i)10-s + 4.95e4·11-s + (−2.25e4 + 6.81e4i)12-s − 4.88e4·13-s + (2.20e4 + 2.17e5i)14-s + (−1.03e5 − 5.96e4i)15-s + (2.41e5 − 1.03e5i)16-s − 5.49e5i·17-s + ⋯
L(s)  = 1  + (0.994 − 0.100i)2-s + (−0.498 + 0.866i)3-s + (0.979 − 0.200i)4-s + 0.609i·5-s + (−0.408 + 0.912i)6-s + 1.51i·7-s + (0.954 − 0.298i)8-s + (−0.503 − 0.864i)9-s + (0.0615 + 0.606i)10-s + 1.02·11-s + (−0.314 + 0.949i)12-s − 0.474·13-s + (0.153 + 1.50i)14-s + (−0.528 − 0.304i)15-s + (0.919 − 0.393i)16-s − 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.314 - 0.949i$
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: \((2,\ 12,\ (\ :9/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.00278 + 1.44677i\)
\(L(\frac12)\) \(\approx\) \(2.00278 + 1.44677i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-22.5 + 2.28i)T \)
3 \( 1 + (69.9 - 121. i)T \)
good5 \( 1 - 852. iT - 1.95e6T^{2} \)
7 \( 1 - 9.64e3iT - 4.03e7T^{2} \)
11 \( 1 - 4.95e4T + 2.35e9T^{2} \)
13 \( 1 + 4.88e4T + 1.06e10T^{2} \)
17 \( 1 + 5.49e5iT - 1.18e11T^{2} \)
19 \( 1 + 1.86e5iT - 3.22e11T^{2} \)
23 \( 1 + 4.90e5T + 1.80e12T^{2} \)
29 \( 1 - 2.90e5iT - 1.45e13T^{2} \)
31 \( 1 - 3.55e6iT - 2.64e13T^{2} \)
37 \( 1 - 1.53e7T + 1.29e14T^{2} \)
41 \( 1 - 2.10e7iT - 3.27e14T^{2} \)
43 \( 1 + 1.45e7iT - 5.02e14T^{2} \)
47 \( 1 + 1.89e7T + 1.11e15T^{2} \)
53 \( 1 + 8.87e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.48e8T + 8.66e15T^{2} \)
61 \( 1 + 8.37e7T + 1.16e16T^{2} \)
67 \( 1 + 1.70e8iT - 2.72e16T^{2} \)
71 \( 1 + 9.80e7T + 4.58e16T^{2} \)
73 \( 1 - 1.94e8T + 5.88e16T^{2} \)
79 \( 1 - 4.25e8iT - 1.19e17T^{2} \)
83 \( 1 - 2.50e8T + 1.86e17T^{2} \)
89 \( 1 + 3.86e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.59e8T + 7.60e17T^{2} \)
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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

−18.27561177235213770260259254025, −16.43093086222128315025355609997, −15.26633371526092571671623145695, −14.38586853840997317438696336827, −12.15923890846764152470349559743, −11.30796721465061932673484200556, −9.473816999833546911971410581328, −6.46350886909951610143944176425, −4.97096461807463948314523646693, −2.91085323031347128000539391893, 1.31364289619366353486472033827, 4.25786331898170463842353834500, 6.27195841331677116836862013042, 7.69741668576881287400661292083, 10.75217343193803207872351074383, 12.24904064857345592039161839214, 13.30275681397609154089128320312, 14.48496695664334777529080248857, 16.70536786643427718971755813453, 17.13683511978724459168406537624

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