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Properties

Label 2-420-105.2-c1-0-12
Degree $2$
Conductor $420$
Sign $0.338 + 0.940i$
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  + (0.591 − 1.62i)3-s + (1.92 + 1.13i)5-s + (−0.225 − 2.63i)7-s + (−2.30 − 1.92i)9-s + (−1.86 + 1.07i)11-s + (3.89 − 3.89i)13-s + (2.98 − 2.46i)15-s + (4.10 + 1.09i)17-s + (−0.631 − 0.364i)19-s + (−4.42 − 1.19i)21-s + (−5.55 + 1.48i)23-s + (2.42 + 4.37i)25-s + (−4.49 + 2.60i)27-s + 3.95·29-s + (−2.33 − 4.04i)31-s + ⋯
L(s)  = 1  + (0.341 − 0.939i)3-s + (0.861 + 0.507i)5-s + (−0.0853 − 0.996i)7-s + (−0.767 − 0.641i)9-s + (−0.561 + 0.324i)11-s + (1.08 − 1.08i)13-s + (0.771 − 0.636i)15-s + (0.994 + 0.266i)17-s + (−0.144 − 0.0835i)19-s + (−0.965 − 0.259i)21-s + (−1.15 + 0.310i)23-s + (0.484 + 0.874i)25-s + (−0.864 + 0.502i)27-s + 0.735·29-s + (−0.418 − 0.725i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.338 + 0.940i$
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: \((2,\ 420,\ (\ :1/2),\ 0.338 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39310 - 0.979251i\)
\(L(\frac12)\) \(\approx\) \(1.39310 - 0.979251i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.591 + 1.62i)T \)
5 \( 1 + (-1.92 - 1.13i)T \)
7 \( 1 + (0.225 + 2.63i)T \)
good11 \( 1 + (1.86 - 1.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.89 + 3.89i)T - 13iT^{2} \)
17 \( 1 + (-4.10 - 1.09i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.631 + 0.364i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.55 - 1.48i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.95T + 29T^{2} \)
31 \( 1 + (2.33 + 4.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.23 + 0.598i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.95iT - 41T^{2} \)
43 \( 1 + (-3.57 + 3.57i)T - 43iT^{2} \)
47 \( 1 + (-2.50 - 9.34i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.66 - 6.22i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.01 - 6.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.20 - 10.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.162 + 0.606i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.00iT - 71T^{2} \)
73 \( 1 + (10.4 + 2.79i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.61 - 3.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.56 - 6.56i)T + 83iT^{2} \)
89 \( 1 + (0.656 - 1.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.0 - 10.0i)T + 97iT^{2} \)
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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

−10.74944339744144037710991070561, −10.32042032688300395479578153344, −9.214849414913830302425145818726, −7.944667371639757214102084191124, −7.46860087941205501015091339822, −6.26379904417566210856171558869, −5.64534379321741858435448348890, −3.77227389216193218443117913788, −2.64561678568666033022447845019, −1.19237055034842073794392452681, 2.00995840606772627655025594537, 3.28008308045430886457942913672, 4.63909503044418224300135283047, 5.58605796558423557181662362717, 6.30577460709574175679079903980, 8.176618415199902531496970825456, 8.730644052813059328477275560108, 9.562155835980322168038160944168, 10.23663454817758606730796090712, 11.29358983163386389029113726786

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