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Properties

Label 2-420-105.2-c1-0-10
Degree $2$
Conductor $420$
Sign $0.998 - 0.0586i$
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 − 0.338i)3-s + (2.15 + 0.592i)5-s + (−2.24 + 1.40i)7-s + (2.77 − 1.15i)9-s + (4.22 − 2.43i)11-s + (−2.53 + 2.53i)13-s + (3.86 + 0.276i)15-s + (−3.56 − 0.955i)17-s + (3.75 + 2.17i)19-s + (−3.33 + 3.14i)21-s + (1.67 − 0.447i)23-s + (4.29 + 2.55i)25-s + (4.31 − 2.89i)27-s − 8.23·29-s + (−4.48 − 7.77i)31-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)3-s + (0.964 + 0.265i)5-s + (−0.847 + 0.531i)7-s + (0.923 − 0.383i)9-s + (1.27 − 0.734i)11-s + (−0.702 + 0.702i)13-s + (0.997 + 0.0713i)15-s + (−0.864 − 0.231i)17-s + (0.862 + 0.497i)19-s + (−0.726 + 0.686i)21-s + (0.348 − 0.0933i)23-s + (0.859 + 0.511i)25-s + (0.830 − 0.556i)27-s − 1.52·29-s + (−0.806 − 1.39i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.998 - 0.0586i$
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: \((2,\ 420,\ (\ :1/2),\ 0.998 - 0.0586i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09426 + 0.0614221i\)
\(L(\frac12)\) \(\approx\) \(2.09426 + 0.0614221i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 0.338i)T \)
5 \( 1 + (-2.15 - 0.592i)T \)
7 \( 1 + (2.24 - 1.40i)T \)
good11 \( 1 + (-4.22 + 2.43i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.53 - 2.53i)T - 13iT^{2} \)
17 \( 1 + (3.56 + 0.955i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.75 - 2.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.67 + 0.447i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.23T + 29T^{2} \)
31 \( 1 + (4.48 + 7.77i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.80 - 0.483i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.33iT - 41T^{2} \)
43 \( 1 + (-0.403 + 0.403i)T - 43iT^{2} \)
47 \( 1 + (-0.106 - 0.397i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.150 + 0.562i)T + (-45.8 - 26.5i)T^{2} \)
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 \( 1 + (3.04 - 11.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 15.2iT - 71T^{2} \)
73 \( 1 + (7.18 + 1.92i)T + (63.2 + 36.5i)T^{2} \)
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

−11.25077316376266470692774030403, −9.875072932644348247257455711198, −9.284292228599078343668338236024, −8.895740475611103321854816951372, −7.35966358908299335235165992661, −6.58740953448283428101663839220, −5.68722932724477110698491932117, −4.01250716659040313489249130359, −2.91739098486022621922271641839, −1.80197849587269848326899480403, 1.65656048726456817098623590627, 3.00615847770833171604584239805, 4.14266238700510480530447036718, 5.32608799648322922101568730829, 6.77762646339557024772131085163, 7.29850942696533750910067911994, 8.803270518763288815091365935609, 9.426874466239633943528640500546, 9.931533479400266888145869211867, 10.92368856082607127057759569009

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