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Properties

Label 2-420-21.5-c1-0-9
Degree $2$
Conductor $420$
Sign $-0.522 + 0.852i$
Analytic cond. $3.35371$
Root an. cond. $1.83131$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.546 − 1.64i)3-s + (−0.5 + 0.866i)5-s + (−1.08 − 2.41i)7-s + (−2.40 − 1.79i)9-s + (−1.17 + 0.675i)11-s − 4.94i·13-s + (1.15 + 1.29i)15-s + (−2.87 − 4.97i)17-s + (2.84 + 1.64i)19-s + (−4.55 + 0.460i)21-s + (4.33 + 2.50i)23-s + (−0.499 − 0.866i)25-s + (−4.26 + 2.96i)27-s − 5.68i·29-s + (−2.45 + 1.41i)31-s + ⋯
L(s)  = 1  + (0.315 − 0.948i)3-s + (−0.223 + 0.387i)5-s + (−0.409 − 0.912i)7-s + (−0.801 − 0.598i)9-s + (−0.353 + 0.203i)11-s − 1.37i·13-s + (0.297 + 0.334i)15-s + (−0.696 − 1.20i)17-s + (0.653 + 0.377i)19-s + (−0.994 + 0.100i)21-s + (0.903 + 0.521i)23-s + (−0.0999 − 0.173i)25-s + (−0.820 + 0.571i)27-s − 1.05i·29-s + (−0.440 + 0.254i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561343 - 1.00236i\)
\(L(\frac12)\) \(\approx\) \(0.561343 - 1.00236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.546 + 1.64i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.08 + 2.41i)T \)
good11 \( 1 + (1.17 - 0.675i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.94iT - 13T^{2} \)
17 \( 1 + (2.87 + 4.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.84 - 1.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.33 - 2.50i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.68iT - 29T^{2} \)
31 \( 1 + (2.45 - 1.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.92 - 3.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 - 4.06T + 43T^{2} \)
47 \( 1 + (-2.84 + 4.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.26 - 0.730i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.34 - 7.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.65 - 0.954i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.51 + 4.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.38iT - 71T^{2} \)
73 \( 1 + (-14.1 + 8.16i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.41 - 4.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + (-8.08 + 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−10.94855585552934434885546758005, −10.02953530087998434446404594857, −9.023456236312968866998054667445, −7.70352295490533108488900032389, −7.41840284947319469017563629144, −6.39529864780199038220743077743, −5.19036453046945821412224502185, −3.56630448789560998640302476466, −2.63546258372083673995536570098, −0.70535248992444532553901501291, 2.28189571598520037390043762602, 3.58071457738901463094670176770, 4.64806646306666560502426355541, 5.59479039296007120733089348160, 6.74928203821112442163075688725, 8.162502585369011838242353699834, 9.068967194661483997624002208808, 9.323241646734890299880918232018, 10.69107184645905680685704087867, 11.30226666208354089606847658727

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