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Properties

Label 2-6-3.2-c12-0-2
Degree $2$
Conductor $6$
Sign $0.848 + 0.528i$
Analytic cond. $5.48396$
Root an. cond. $2.34178$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 45.2i·2-s + (385. − 618. i)3-s − 2.04e3·4-s − 1.63e4i·5-s + (2.79e4 + 1.74e4i)6-s + 2.13e5·7-s − 9.26e4i·8-s + (−2.34e5 − 4.77e5i)9-s + 7.39e5·10-s − 852. i·11-s + (−7.89e5 + 1.26e6i)12-s − 3.33e6·13-s + 9.64e6i·14-s + (−1.01e7 − 6.30e6i)15-s + 4.19e6·16-s − 4.86e6i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.528 − 0.848i)3-s − 0.500·4-s − 1.04i·5-s + (0.600 + 0.374i)6-s + 1.81·7-s − 0.353i·8-s + (−0.440 − 0.897i)9-s + 0.739·10-s − 0.000481i·11-s + (−0.264 + 0.424i)12-s − 0.690·13-s + 1.28i·14-s + (−0.887 − 0.553i)15-s + 0.250·16-s − 0.201i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.848 + 0.528i$
Analytic conductor: \(5.48396\)
Root analytic conductor: \(2.34178\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :6),\ 0.848 + 0.528i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.82152 - 0.521162i\)
\(L(\frac12)\) \(\approx\) \(1.82152 - 0.521162i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 45.2iT \)
3 \( 1 + (-385. + 618. i)T \)
good5 \( 1 + 1.63e4iT - 2.44e8T^{2} \)
7 \( 1 - 2.13e5T + 1.38e10T^{2} \)
11 \( 1 + 852. iT - 3.13e12T^{2} \)
13 \( 1 + 3.33e6T + 2.32e13T^{2} \)
17 \( 1 + 4.86e6iT - 5.82e14T^{2} \)
19 \( 1 - 1.39e5T + 2.21e15T^{2} \)
23 \( 1 - 2.14e8iT - 2.19e16T^{2} \)
29 \( 1 - 8.70e8iT - 3.53e17T^{2} \)
31 \( 1 - 1.12e9T + 7.87e17T^{2} \)
37 \( 1 - 1.03e8T + 6.58e18T^{2} \)
41 \( 1 - 6.26e8iT - 2.25e19T^{2} \)
43 \( 1 + 5.27e9T + 3.99e19T^{2} \)
47 \( 1 + 1.16e10iT - 1.16e20T^{2} \)
53 \( 1 - 2.31e10iT - 4.91e20T^{2} \)
59 \( 1 - 3.43e10iT - 1.77e21T^{2} \)
61 \( 1 + 1.42e10T + 2.65e21T^{2} \)
67 \( 1 + 4.78e10T + 8.18e21T^{2} \)
71 \( 1 + 2.46e11iT - 1.64e22T^{2} \)
73 \( 1 - 5.45e9T + 2.29e22T^{2} \)
79 \( 1 + 3.27e9T + 5.90e22T^{2} \)
83 \( 1 - 1.55e11iT - 1.06e23T^{2} \)
89 \( 1 + 2.37e10iT - 2.46e23T^{2} \)
97 \( 1 - 7.71e11T + 6.93e23T^{2} \)
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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

−19.96614783775376513956433929054, −18.11526957541138320467464301455, −17.09008960730660097679570296307, −14.97607147414274186435781254509, −13.69761656897018926542940204989, −12.01373552596451827381966293520, −8.790394794587510301183306555561, −7.61431397023134275482323105549, −5.02441812133786742099649360934, −1.33983742148211854761494666971, 2.43520548979333277248717840343, 4.57867095536885444855804477088, 8.179749050940782613802923553780, 10.27260477188170514515708973493, 11.42638366898050964808187764095, 14.17193156421105519318947966466, 14.95709231531537142669963541456, 17.41273698192477745823493911977, 18.90188626600168903086708429505, 20.52007013971099503107923404373

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