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Properties

Label 2-420-105.2-c1-0-13
Degree $2$
Conductor $420$
Sign $0.538 + 0.842i$
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  + (1.68 − 0.396i)3-s + (0.133 − 2.23i)5-s + (1.78 − 1.94i)7-s + (2.68 − 1.33i)9-s + (−2.00 + 1.15i)11-s + (−2 + 2i)13-s + (−0.658 − 3.81i)15-s + (−3.16 − 0.847i)17-s + (−0.274 − 0.158i)19-s + (2.24 − 3.99i)21-s + (1.58 − 0.423i)23-s + (−4.96 − 0.598i)25-s + (4 − 3.31i)27-s + 7.31·29-s + (3 + 5.19i)31-s + ⋯
L(s)  = 1  + (0.973 − 0.228i)3-s + (0.0599 − 0.998i)5-s + (0.676 − 0.736i)7-s + (0.895 − 0.445i)9-s + (−0.604 + 0.349i)11-s + (−0.554 + 0.554i)13-s + (−0.169 − 0.985i)15-s + (−0.767 − 0.205i)17-s + (−0.0629 − 0.0363i)19-s + (0.490 − 0.871i)21-s + (0.329 − 0.0884i)23-s + (−0.992 − 0.119i)25-s + (0.769 − 0.638i)27-s + 1.35·29-s + (0.538 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.538 + 0.842i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.538 + 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70281 - 0.933036i\)
\(L(\frac12)\) \(\approx\) \(1.70281 - 0.933036i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.396i)T \)
5 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 + (-1.78 + 1.94i)T \)
good11 \( 1 + (2.00 - 1.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (3.16 + 0.847i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.274 + 0.158i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.58 + 0.423i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.31T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.46 + 1.46i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 9.63iT - 41T^{2} \)
43 \( 1 + (-1.84 + 1.84i)T - 43iT^{2} \)
47 \( 1 + (-1.83 - 6.83i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.67 - 9.99i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.15 + 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.65 - 8.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.62 + 9.77i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.31iT - 71T^{2} \)
73 \( 1 + (-9.06 - 2.42i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.92 + 2.84i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.84 + 3.84i)T + 83iT^{2} \)
89 \( 1 + (8.29 - 14.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.31 + 9.31i)T + 97iT^{2} \)
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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

−10.99970774642663482221711241177, −9.938085584728389913164777600823, −9.156685131617391514192503561741, −8.254367586701064398904075459524, −7.60705442018236130310189340025, −6.56916849696238470930666712463, −4.79262384026704413911529518372, −4.37198907736516681311598743184, −2.65450022819414717950104152951, −1.30881389759979337209156216936, 2.25683003652281062478416199815, 2.95428473663836291884916885180, 4.36687308441370996377342711481, 5.56162355292054663779437413030, 6.82594295258994099885012555461, 7.86504170625875804198705964114, 8.463224323752086527386081066470, 9.554439997019523485345377221847, 10.42039140382465521627201354104, 11.13266464575445632818664581864

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