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Properties

Label 2-420-105.23-c1-0-5
Degree $2$
Conductor $420$
Sign $-0.0116 - 0.999i$
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  + (0.338 + 1.69i)3-s + (1.59 + 1.57i)5-s + (1.40 + 2.24i)7-s + (−2.77 + 1.15i)9-s + (4.22 − 2.43i)11-s + (−2.53 − 2.53i)13-s + (−2.12 + 3.23i)15-s + (−0.955 + 3.56i)17-s + (−3.75 − 2.17i)19-s + (−3.33 + 3.14i)21-s + (0.447 + 1.67i)23-s + (0.0649 + 4.99i)25-s + (−2.89 − 4.31i)27-s + 8.23·29-s + (−4.48 − 7.77i)31-s + ⋯
L(s)  = 1  + (0.195 + 0.980i)3-s + (0.711 + 0.702i)5-s + (0.531 + 0.847i)7-s + (−0.923 + 0.383i)9-s + (1.27 − 0.734i)11-s + (−0.702 − 0.702i)13-s + (−0.549 + 0.835i)15-s + (−0.231 + 0.864i)17-s + (−0.862 − 0.497i)19-s + (−0.726 + 0.686i)21-s + (0.0933 + 0.348i)23-s + (0.0129 + 0.999i)25-s + (−0.556 − 0.830i)27-s + 1.52·29-s + (−0.806 − 1.39i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0116 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0116 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0116 - 0.999i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.0116 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18178 + 1.19565i\)
\(L(\frac12)\) \(\approx\) \(1.18178 + 1.19565i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.338 - 1.69i)T \)
5 \( 1 + (-1.59 - 1.57i)T \)
7 \( 1 + (-1.40 - 2.24i)T \)
good11 \( 1 + (-4.22 + 2.43i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.53 + 2.53i)T + 13iT^{2} \)
17 \( 1 + (0.955 - 3.56i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.75 + 2.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.447 - 1.67i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.23T + 29T^{2} \)
31 \( 1 + (4.48 + 7.77i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.483 - 1.80i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.33iT - 41T^{2} \)
43 \( 1 + (-0.403 - 0.403i)T + 43iT^{2} \)
47 \( 1 + (-0.397 + 0.106i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.562 - 0.150i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.59 + 2.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.05 - 3.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.3 - 3.04i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 15.2iT - 71T^{2} \)
73 \( 1 + (-1.92 + 7.18i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.84 + 1.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \)
89 \( 1 + (-6.60 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.44 - 1.44i)T - 97iT^{2} \)
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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

−11.18889077955540581850298952742, −10.56415748277872430380452526384, −9.564631759251616396725993155859, −8.900558477786344941815035378535, −8.008091005686643032532907277219, −6.42375117719725044562414142635, −5.74667925630658099255227355663, −4.60370959428541096030153832988, −3.31916847189892851960033271928, −2.20548363300569337281712755275, 1.19805017009623957468899539809, 2.22923817073408861518415485749, 4.14672166341648466690265483001, 5.12287395137036356018295155781, 6.62161548469940686318529313160, 7.00060542842602021071251485584, 8.267887945442180060858340661489, 9.061042589362814967435910785836, 9.908301389476914999218050433614, 11.09720921812688250254280056205

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