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Properties

Label 2-420-105.89-c1-0-7
Degree $2$
Conductor $420$
Sign $0.667 + 0.744i$
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  + (−1.44 + 0.956i)3-s + (−2.00 + 0.980i)5-s + (0.477 − 2.60i)7-s + (1.16 − 2.76i)9-s + (1.34 − 0.773i)11-s − 4.18·13-s + (1.96 − 3.33i)15-s + (4.42 − 2.55i)17-s + (4.62 + 2.67i)19-s + (1.80 + 4.21i)21-s + (1.82 − 3.15i)23-s + (3.07 − 3.94i)25-s + (0.954 + 5.10i)27-s − 9.79i·29-s + (6.79 − 3.92i)31-s + ⋯
L(s)  = 1  + (−0.833 + 0.552i)3-s + (−0.898 + 0.438i)5-s + (0.180 − 0.983i)7-s + (0.389 − 0.920i)9-s + (0.404 − 0.233i)11-s − 1.16·13-s + (0.506 − 0.862i)15-s + (1.07 − 0.619i)17-s + (1.06 + 0.613i)19-s + (0.392 + 0.919i)21-s + (0.380 − 0.658i)23-s + (0.615 − 0.788i)25-s + (0.183 + 0.982i)27-s − 1.81i·29-s + (1.22 − 0.705i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.717312 - 0.320192i\)
\(L(\frac12)\) \(\approx\) \(0.717312 - 0.320192i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 - 0.956i)T \)
5 \( 1 + (2.00 - 0.980i)T \)
7 \( 1 + (-0.477 + 2.60i)T \)
good11 \( 1 + (-1.34 + 0.773i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.18T + 13T^{2} \)
17 \( 1 + (-4.42 + 2.55i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.62 - 2.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.82 + 3.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.79iT - 29T^{2} \)
31 \( 1 + (-6.79 + 3.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.96 + 1.71i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.82T + 41T^{2} \)
43 \( 1 - 3.79iT - 43T^{2} \)
47 \( 1 + (2.16 + 1.24i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.395 - 0.684i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.73 + 4.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.76 + 3.90i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.67 + 5.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.97iT - 71T^{2} \)
73 \( 1 + (-6.19 - 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.12 - 1.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.26iT - 83T^{2} \)
89 \( 1 + (-7.65 + 13.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−11.23048049835473845201711257150, −10.05198311516892469930476787645, −9.792866662920822472881941169350, −8.078542033815379264851430669144, −7.33062085989856822917565592869, −6.43745928525567335748595743701, −5.09646600624601272734892365909, −4.22223260148376681416448191808, −3.21984029999759986002410762033, −0.63338399548691693541691643541, 1.39502375522230718122585144573, 3.17980185829810227436067597789, 4.92214106925989680385334089419, 5.30877883644371163762239670808, 6.75865415919270523348979625509, 7.52890185976274781217733196300, 8.443582228998437816174480503268, 9.485102139507229169887950698636, 10.59718216499585798268162534893, 11.70015517487720656386225534070

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