Properties

Label 2-420-21.5-c1-0-6
Degree $2$
Conductor $420$
Sign $0.908 + 0.418i$
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  + (1.69 + 0.348i)3-s + (0.5 − 0.866i)5-s + (−1.08 − 2.41i)7-s + (2.75 + 1.18i)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 + (−0.993 − 4.47i)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  + (0.979 + 0.201i)3-s + (0.223 − 0.387i)5-s + (−0.409 − 0.912i)7-s + (0.918 + 0.394i)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 + (−0.216 − 0.976i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.908 + 0.418i$
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: \((2,\ 420,\ (\ :1/2),\ 0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90211 - 0.417139i\)
\(L(\frac12)\) \(\approx\) \(1.90211 - 0.417139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.348i)T \)
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

−10.71810641736030740973594400342, −10.20087137047525077299336940217, −9.394739909854725833746658497891, −8.257589799409971328307467980679, −7.75340171052336542267799108022, −6.50832843635172370000150521511, −5.27021577079066714924897151923, −3.92241844442300222375718619344, −3.17668275280752308422342574141, −1.38338971620188910431068467300, 1.93016579008400675698332641412, 2.95967889714600831243729318901, 4.14408008463203194631867713191, 5.61789532050222591584695961825, 6.77252705426223268870258159436, 7.48631348197818731712084686246, 8.699129510537987759605897543702, 9.493943977346472064071827046075, 9.878708520269529664795046724642, 11.57049967683239618473218476086

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