Properties

Label 2-420-105.23-c1-0-12
Degree $2$
Conductor $420$
Sign $-0.786 + 0.617i$
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  + (−1.14 + 1.30i)3-s + (−1.59 − 1.57i)5-s + (1.40 + 2.24i)7-s + (−0.388 − 2.97i)9-s + (−4.22 + 2.43i)11-s + (−2.53 − 2.53i)13-s + (3.86 − 0.276i)15-s + (0.955 − 3.56i)17-s + (−3.75 − 2.17i)19-s + (−4.52 − 0.731i)21-s + (−0.447 − 1.67i)23-s + (0.0649 + 4.99i)25-s + (4.31 + 2.89i)27-s − 8.23·29-s + (−4.48 − 7.77i)31-s + ⋯
L(s)  = 1  + (−0.659 + 0.751i)3-s + (−0.711 − 0.702i)5-s + (0.531 + 0.847i)7-s + (−0.129 − 0.991i)9-s + (−1.27 + 0.734i)11-s + (−0.702 − 0.702i)13-s + (0.997 − 0.0713i)15-s + (0.231 − 0.864i)17-s + (−0.862 − 0.497i)19-s + (−0.987 − 0.159i)21-s + (−0.0933 − 0.348i)23-s + (0.0129 + 0.999i)25-s + (0.830 + 0.556i)27-s − 1.52·29-s + (−0.806 − 1.39i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.786 + 0.617i$
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: \((2,\ 420,\ (\ :1/2),\ -0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0346365 - 0.100298i\)
\(L(\frac12)\) \(\approx\) \(0.0346365 - 0.100298i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 1.30i)T \)
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

−11.00304866337763116822690626947, −9.918961845030832597174573568051, −9.142165912415648999839794475318, −8.109218970392650976738974915448, −7.26430242227077094113095809594, −5.55556984159992568315252681619, −5.11487005038438194282937698111, −4.16665202488619069614608348471, −2.53257189224792287913328274895, −0.06946519777801232946451514848, 1.95965388351204761540543686743, 3.58055392752335129331534180819, 4.84838450108933651319716463811, 5.97632836379287184380780602518, 7.07988794146257739455093276513, 7.68055251972586344318620347713, 8.413903669628155110659472270045, 10.19012912954027691393194107992, 10.88539227517083609789110268890, 11.31323006732944648166830299504

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