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Properties

Label 2-420-21.5-c1-0-5
Degree $2$
Conductor $420$
Sign $-0.133 + 0.991i$
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  + (−1.63 − 0.576i)3-s + (0.5 − 0.866i)5-s + (−1.73 + 1.99i)7-s + (2.33 + 1.88i)9-s + (3.38 − 1.95i)11-s − 6.06i·13-s + (−1.31 + 1.12i)15-s + (−1.53 − 2.65i)17-s + (−2.94 − 1.70i)19-s + (3.98 − 2.26i)21-s + (−2.48 − 1.43i)23-s + (−0.499 − 0.866i)25-s + (−2.72 − 4.42i)27-s − 7.97i·29-s + (−5.63 + 3.25i)31-s + ⋯
L(s)  = 1  + (−0.942 − 0.333i)3-s + (0.223 − 0.387i)5-s + (−0.655 + 0.755i)7-s + (0.778 + 0.628i)9-s + (1.01 − 0.588i)11-s − 1.68i·13-s + (−0.339 + 0.290i)15-s + (−0.371 − 0.643i)17-s + (−0.676 − 0.390i)19-s + (0.869 − 0.493i)21-s + (−0.518 − 0.299i)23-s + (−0.0999 − 0.173i)25-s + (−0.524 − 0.851i)27-s − 1.48i·29-s + (−1.01 + 0.583i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.133 + 0.991i$
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: \((2,\ 420,\ (\ :1/2),\ -0.133 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.547246 - 0.625933i\)
\(L(\frac12)\) \(\approx\) \(0.547246 - 0.625933i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 + 0.576i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.73 - 1.99i)T \)
good11 \( 1 + (-3.38 + 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.06iT - 13T^{2} \)
17 \( 1 + (1.53 + 2.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.94 + 1.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.48 + 1.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.97iT - 29T^{2} \)
31 \( 1 + (5.63 - 3.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0654 - 0.113i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 - 4.43T + 43T^{2} \)
47 \( 1 + (-5.02 + 8.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.64 - 2.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.28 + 2.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.44 - 4.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.99 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.63iT - 71T^{2} \)
73 \( 1 + (6.72 - 3.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.22 - 2.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.63T + 83T^{2} \)
89 \( 1 + (4.11 - 7.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.74iT - 97T^{2} \)
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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

−11.07291982912799558007537727497, −10.11333235625076391523753673147, −9.190289538949353639235817066481, −8.236171505784303895726959331428, −7.01366777326298488003042460485, −5.97825883050246931678399589495, −5.53272812592143533759997290617, −4.12469961719517472602044917808, −2.50205350593252423794052630243, −0.61647495490232449492829823577, 1.68501250288476466254959133600, 3.86127421000909461890691022396, 4.36885122459410731615936255138, 6.00418445876864772472489007554, 6.63405807539164586292931848803, 7.34054669711306333325064563211, 9.180264955083937885875722708281, 9.605501684988352789549246231799, 10.71945604681328626383018214679, 11.21241980918754727544411127710

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