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Properties

Label 2-12-12.11-c11-0-0
Degree $2$
Conductor $12$
Sign $-0.985 - 0.170i$
Analytic cond. $9.22011$
Root an. cond. $3.03646$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−32.0 + 31.9i)2-s + (70.6 − 414. i)3-s + (5.02 − 2.04e3i)4-s + 1.84e3i·5-s + (1.09e4 + 1.55e4i)6-s + 1.63e4i·7-s + (6.52e4 + 6.57e4i)8-s + (−1.67e5 − 5.86e4i)9-s + (−5.88e4 − 5.90e4i)10-s − 4.85e5·11-s + (−8.49e5 − 1.46e5i)12-s − 1.36e6·13-s + (−5.21e5 − 5.22e5i)14-s + (7.64e5 + 1.30e5i)15-s + (−4.19e6 − 2.05e4i)16-s + 8.90e6i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.706i)2-s + (0.167 − 0.985i)3-s + (0.00245 − 0.999i)4-s + 0.263i·5-s + (0.577 + 0.816i)6-s + 0.366i·7-s + (0.704 + 0.709i)8-s + (−0.943 − 0.330i)9-s + (−0.186 − 0.186i)10-s − 0.909·11-s + (−0.985 − 0.170i)12-s − 1.02·13-s + (−0.258 − 0.259i)14-s + (0.259 + 0.0442i)15-s + (−0.999 − 0.00490i)16-s + 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.985 - 0.170i$
Analytic conductor: \(9.22011\)
Root analytic conductor: \(3.03646\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :11/2),\ -0.985 - 0.170i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0104821 + 0.122222i\)
\(L(\frac12)\) \(\approx\) \(0.0104821 + 0.122222i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (32.0 - 31.9i)T \)
3 \( 1 + (-70.6 + 414. i)T \)
good5 \( 1 - 1.84e3iT - 4.88e7T^{2} \)
7 \( 1 - 1.63e4iT - 1.97e9T^{2} \)
11 \( 1 + 4.85e5T + 2.85e11T^{2} \)
13 \( 1 + 1.36e6T + 1.79e12T^{2} \)
17 \( 1 - 8.90e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.07e5iT - 1.16e14T^{2} \)
23 \( 1 + 5.58e7T + 9.52e14T^{2} \)
29 \( 1 - 1.27e8iT - 1.22e16T^{2} \)
31 \( 1 + 1.30e8iT - 2.54e16T^{2} \)
37 \( 1 + 3.44e8T + 1.77e17T^{2} \)
41 \( 1 + 9.40e8iT - 5.50e17T^{2} \)
43 \( 1 + 9.39e8iT - 9.29e17T^{2} \)
47 \( 1 - 9.85e8T + 2.47e18T^{2} \)
53 \( 1 + 1.84e9iT - 9.26e18T^{2} \)
59 \( 1 + 5.76e9T + 3.01e19T^{2} \)
61 \( 1 + 7.38e9T + 4.35e19T^{2} \)
67 \( 1 - 1.05e10iT - 1.22e20T^{2} \)
71 \( 1 + 1.58e10T + 2.31e20T^{2} \)
73 \( 1 - 1.99e10T + 3.13e20T^{2} \)
79 \( 1 - 1.66e10iT - 7.47e20T^{2} \)
83 \( 1 + 3.12e10T + 1.28e21T^{2} \)
89 \( 1 + 5.55e10iT - 2.77e21T^{2} \)
97 \( 1 + 4.52e10T + 7.15e21T^{2} \)
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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

−18.13792289351838258583351307960, −17.00676253811959893774968056989, −15.29870182277368224955886817908, −14.11399265661162772797070107938, −12.42672161986344896206307327416, −10.45417102390122566124686912586, −8.552770844701436245256973768801, −7.30926770065966409480902752670, −5.75651043829329145545759670385, −2.07419235510862358350900709215, 0.06858613459222260317586510183, 2.74056490778573808201231503474, 4.65654604026916315307638720725, 7.82600434005648478709778683333, 9.452736840213270404345056230202, 10.48934056856502073162008878882, 12.01227957946228185619307486499, 13.84276356754057071119409423665, 15.79362611646079927878423701356, 16.80121517970552580892007769520

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