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Properties

Label 2-6-3.2-c14-0-1
Degree $2$
Conductor $6$
Sign $0.379 - 0.925i$
Analytic cond. $7.45973$
Root an. cond. $2.73125$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 90.5i·2-s + (−2.02e3 − 829. i)3-s − 8.19e3·4-s + 4.04e4i·5-s + (−7.50e4 + 1.83e5i)6-s + 3.88e5·7-s + 7.41e5i·8-s + (3.40e6 + 3.35e6i)9-s + 3.65e6·10-s + 3.20e7i·11-s + (1.65e7 + 6.79e6i)12-s − 5.64e7·13-s − 3.51e7i·14-s + (3.35e7 − 8.18e7i)15-s + 6.71e7·16-s − 2.36e8i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.925 − 0.379i)3-s − 0.500·4-s + 0.517i·5-s + (−0.268 + 0.654i)6-s + 0.472·7-s + 0.353i·8-s + (0.712 + 0.701i)9-s + 0.365·10-s + 1.64i·11-s + (0.462 + 0.189i)12-s − 0.899·13-s − 0.333i·14-s + (0.196 − 0.478i)15-s + 0.250·16-s − 0.576i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(7.45973\)
Root analytic conductor: \(2.73125\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :7),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.554774 + 0.372263i\)
\(L(\frac12)\) \(\approx\) \(0.554774 + 0.372263i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 90.5iT \)
3 \( 1 + (2.02e3 + 829. i)T \)
good5 \( 1 - 4.04e4iT - 6.10e9T^{2} \)
7 \( 1 - 3.88e5T + 6.78e11T^{2} \)
11 \( 1 - 3.20e7iT - 3.79e14T^{2} \)
13 \( 1 + 5.64e7T + 3.93e15T^{2} \)
17 \( 1 + 2.36e8iT - 1.68e17T^{2} \)
19 \( 1 + 1.43e9T + 7.99e17T^{2} \)
23 \( 1 - 6.74e9iT - 1.15e19T^{2} \)
29 \( 1 - 8.00e9iT - 2.97e20T^{2} \)
31 \( 1 + 7.26e9T + 7.56e20T^{2} \)
37 \( 1 - 2.67e10T + 9.01e21T^{2} \)
41 \( 1 - 1.45e11iT - 3.79e22T^{2} \)
43 \( 1 + 1.21e11T + 7.38e22T^{2} \)
47 \( 1 + 5.48e11iT - 2.56e23T^{2} \)
53 \( 1 + 8.99e11iT - 1.37e24T^{2} \)
59 \( 1 - 1.29e12iT - 6.19e24T^{2} \)
61 \( 1 - 1.73e12T + 9.87e24T^{2} \)
67 \( 1 + 7.50e12T + 3.67e25T^{2} \)
71 \( 1 - 4.94e12iT - 8.27e25T^{2} \)
73 \( 1 - 9.76e12T + 1.22e26T^{2} \)
79 \( 1 + 2.22e13T + 3.68e26T^{2} \)
83 \( 1 - 2.67e13iT - 7.36e26T^{2} \)
89 \( 1 + 7.12e13iT - 1.95e27T^{2} \)
97 \( 1 - 8.75e13T + 6.52e27T^{2} \)
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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

−19.68539970800454246324316672915, −18.18453078972595517878293388017, −17.23539814415511805231280608623, −14.89477518214729857967489705718, −12.83152177786734421431886419335, −11.52453331504707407796859907224, −10.04624221969914709509749963130, −7.20382384352986118606052028050, −4.81953122621215324516264381273, −1.90768625393773465407312539786, 0.39751691022558813402349230514, 4.59179123947172697653023921407, 6.20148693632216756392073389180, 8.577000713488446352290373233176, 10.74605607370467889893849412679, 12.63875444110779994299284585108, 14.71070282150924128142247066724, 16.40367785512538643465790090225, 17.16624893212514624982151180213, 18.85831583999379467582968098540

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