Properties

Label 2-420-7.2-c1-0-3
Degree $2$
Conductor $420$
Sign $0.947 + 0.318i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (1.62 + 2.09i)7-s + (−0.499 + 0.866i)9-s + (2.12 + 3.67i)11-s + 3.24·13-s − 0.999·15-s + (−2.12 − 3.67i)17-s + (3.5 − 6.06i)19-s + (0.999 − 2.44i)21-s + (2.12 − 3.67i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s − 1.75·29-s + (4.74 + 8.21i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.612 + 0.790i)7-s + (−0.166 + 0.288i)9-s + (0.639 + 1.10i)11-s + 0.899·13-s − 0.258·15-s + (−0.514 − 0.891i)17-s + (0.802 − 1.39i)19-s + (0.218 − 0.534i)21-s + (0.442 − 0.766i)23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s − 0.326·29-s + (0.851 + 1.47i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44540 - 0.236389i\)
\(L(\frac12)\) \(\approx\) \(1.44540 - 0.236389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.62 - 2.09i)T \)
good11 \( 1 + (-2.12 - 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
17 \( 1 + (2.12 + 3.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.12 + 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.75T + 29T^{2} \)
31 \( 1 + (-4.74 - 8.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.62 - 2.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.24 + 7.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.12 - 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.24 - 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.62 - 4.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (4.62 + 8.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (-5.12 + 8.87i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.485T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.49783617161284007444756744613, −10.28789655868382563704450326831, −9.057994219058298670650212399533, −8.649486418416454542758751085691, −7.27016364787680368618856649709, −6.55111335422242991701834561640, −5.27795176955433360983457336322, −4.58061929628997007073982246359, −2.68503386982822601627751115173, −1.35276220967991546484170318303, 1.36465581120633051088249575763, 3.44622082449789052499283315338, 4.16337751868363385412291349195, 5.66799939797787732853098303803, 6.30159179171425474207427120701, 7.62314133310564403429009010386, 8.508798699354414367742757719756, 9.553347940567516356006897002772, 10.52405546012742115465818457242, 11.14128790751630802422128586241

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