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Properties

Label 2-3-3.2-c20-0-2
Degree $2$
Conductor $3$
Sign $0.959 + 0.281i$
Analytic cond. $7.60541$
Root an. cond. $2.75779$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 966. i·2-s + (5.66e4 + 1.66e4i)3-s + 1.14e5·4-s + 1.69e7i·5-s + (1.60e7 − 5.47e7i)6-s + 2.16e8·7-s − 1.12e9i·8-s + (2.93e9 + 1.88e9i)9-s + 1.63e10·10-s − 1.25e10i·11-s + (6.51e9 + 1.90e9i)12-s − 9.02e10·13-s − 2.09e11i·14-s + (−2.80e11 + 9.57e11i)15-s − 9.65e11·16-s − 5.69e11i·17-s + ⋯
L(s)  = 1  − 0.943i·2-s + (0.959 + 0.281i)3-s + 0.109·4-s + 1.73i·5-s + (0.265 − 0.905i)6-s + 0.766·7-s − 1.04i·8-s + (0.841 + 0.540i)9-s + 1.63·10-s − 0.482i·11-s + (0.105 + 0.0308i)12-s − 0.654·13-s − 0.723i·14-s + (−0.487 + 1.66i)15-s − 0.878·16-s − 0.282i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.959 + 0.281i$
Analytic conductor: \(7.60541\)
Root analytic conductor: \(2.75779\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :10),\ 0.959 + 0.281i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(2.55222 - 0.366506i\)
\(L(\frac12)\) \(\approx\) \(2.55222 - 0.366506i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.66e4 - 1.66e4i)T \)
good2 \( 1 + 966. iT - 1.04e6T^{2} \)
5 \( 1 - 1.69e7iT - 9.53e13T^{2} \)
7 \( 1 - 2.16e8T + 7.97e16T^{2} \)
11 \( 1 + 1.25e10iT - 6.72e20T^{2} \)
13 \( 1 + 9.02e10T + 1.90e22T^{2} \)
17 \( 1 + 5.69e11iT - 4.06e24T^{2} \)
19 \( 1 - 4.66e12T + 3.75e25T^{2} \)
23 \( 1 - 1.67e13iT - 1.71e27T^{2} \)
29 \( 1 + 5.50e14iT - 1.76e29T^{2} \)
31 \( 1 + 5.98e14T + 6.71e29T^{2} \)
37 \( 1 + 6.64e15T + 2.31e31T^{2} \)
41 \( 1 - 6.99e15iT - 1.80e32T^{2} \)
43 \( 1 - 1.44e16T + 4.67e32T^{2} \)
47 \( 1 + 9.26e16iT - 2.76e33T^{2} \)
53 \( 1 - 1.20e16iT - 3.05e34T^{2} \)
59 \( 1 - 1.03e17iT - 2.61e35T^{2} \)
61 \( 1 + 3.83e17T + 5.08e35T^{2} \)
67 \( 1 - 1.36e18T + 3.32e36T^{2} \)
71 \( 1 - 7.04e17iT - 1.05e37T^{2} \)
73 \( 1 - 1.09e18T + 1.84e37T^{2} \)
79 \( 1 + 6.22e18T + 8.96e37T^{2} \)
83 \( 1 - 1.97e19iT - 2.40e38T^{2} \)
89 \( 1 + 8.06e18iT - 9.72e38T^{2} \)
97 \( 1 - 5.28e19T + 5.43e39T^{2} \)
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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

−21.13868261458531292024145354107, −19.57876794061439941119529629212, −18.45646867332355154489393703598, −15.33824220814270624252628896887, −14.00144413789072722791623203163, −11.36918682935804761884844038010, −10.04449514753480083370993853457, −7.32756742865891911568767457859, −3.40642993138832112737378445879, −2.17993105286190126971561791800, 1.64550885848364297511903844873, 4.94589826862370225574677285229, 7.61223187642422132809990586230, 8.893942238921179221872692250108, 12.44648552627923266768067677672, 14.36777956959162849009063378589, 15.93476102284430895510705466460, 17.45377528884081053627302588523, 20.02871330881728904628797055474, 20.84714363758827746180983531665

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