Properties

Label 2-12-3.2-c10-0-0
Degree $2$
Conductor $12$
Sign $-0.481 - 0.876i$
Analytic cond. $7.62428$
Root an. cond. $2.76121$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (117 + 212. i)3-s + 1.27e3i·5-s − 1.03e4·7-s + (−3.16e4 + 4.98e4i)9-s + 2.92e5i·11-s − 2.56e5·13-s + (−2.72e5 + 1.49e5i)15-s − 5.57e5i·17-s + 3.19e6·19-s + (−1.20e6 − 2.19e6i)21-s − 8.35e6i·23-s + 8.13e6·25-s + (−1.43e7 − 9.14e5i)27-s + 3.14e7i·29-s + 2.31e7·31-s + ⋯
L(s)  = 1  + (0.481 + 0.876i)3-s + 0.408i·5-s − 0.613·7-s + (−0.536 + 0.843i)9-s + 1.81i·11-s − 0.691·13-s + (−0.358 + 0.196i)15-s − 0.392i·17-s + 1.29·19-s + (−0.295 − 0.538i)21-s − 1.29i·23-s + 0.832·25-s + (−0.997 − 0.0637i)27-s + 1.53i·29-s + 0.808·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(7.62428\)
Root analytic conductor: \(2.76121\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :5),\ -0.481 - 0.876i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.785930 + 1.32846i\)
\(L(\frac12)\) \(\approx\) \(0.785930 + 1.32846i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-117 - 212. i)T \)
good5 \( 1 - 1.27e3iT - 9.76e6T^{2} \)
7 \( 1 + 1.03e4T + 2.82e8T^{2} \)
11 \( 1 - 2.92e5iT - 2.59e10T^{2} \)
13 \( 1 + 2.56e5T + 1.37e11T^{2} \)
17 \( 1 + 5.57e5iT - 2.01e12T^{2} \)
19 \( 1 - 3.19e6T + 6.13e12T^{2} \)
23 \( 1 + 8.35e6iT - 4.14e13T^{2} \)
29 \( 1 - 3.14e7iT - 4.20e14T^{2} \)
31 \( 1 - 2.31e7T + 8.19e14T^{2} \)
37 \( 1 - 2.97e7T + 4.80e15T^{2} \)
41 \( 1 - 9.48e5iT - 1.34e16T^{2} \)
43 \( 1 - 2.47e8T + 2.16e16T^{2} \)
47 \( 1 + 3.29e8iT - 5.25e16T^{2} \)
53 \( 1 - 5.51e8iT - 1.74e17T^{2} \)
59 \( 1 - 3.59e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.05e9T + 7.13e17T^{2} \)
67 \( 1 + 3.61e8T + 1.82e18T^{2} \)
71 \( 1 + 9.31e8iT - 3.25e18T^{2} \)
73 \( 1 - 3.74e8T + 4.29e18T^{2} \)
79 \( 1 + 1.13e8T + 9.46e18T^{2} \)
83 \( 1 - 4.91e9iT - 1.55e19T^{2} \)
89 \( 1 - 3.28e9iT - 3.11e19T^{2} \)
97 \( 1 - 2.80e9T + 7.37e19T^{2} \)
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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

−18.18344807776616438502901074944, −16.57829247418246649747997379673, −15.29688278373390057066468876608, −14.26245249197360428687948904889, −12.40024581846979254152415187300, −10.39490752214157436276564585247, −9.353126669734490231659613157754, −7.22311862131100415177641827797, −4.72998917112269484812631151539, −2.76237648915955229023385816189, 0.789580357794664697254370616052, 3.10796468535629369236409047632, 6.00020131146609420356958428960, 7.85004543163858515396682652204, 9.334061680049647402611668270697, 11.62280422619632808234234081300, 13.07914880803347042607814136231, 14.09078359654878080844594455088, 15.93086516756180354100302406789, 17.34381799898514274396846532110

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