↑

Properties

Label 2-420-105.23-c1-0-1
Degree $2$
Conductor $420$
Sign $-0.427 - 0.903i$
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  + (0.146 − 1.72i)3-s + (−1.16 + 1.91i)5-s + (−1.34 + 2.28i)7-s + (−2.95 − 0.507i)9-s + (−4.85 + 2.80i)11-s + (0.354 + 0.354i)13-s + (3.12 + 2.28i)15-s + (−0.959 + 3.57i)17-s + (−2.40 − 1.39i)19-s + (3.73 + 2.64i)21-s + (−1.06 − 3.95i)23-s + (−2.30 − 4.43i)25-s + (−1.30 + 5.02i)27-s + 8.44·29-s + (−0.730 − 1.26i)31-s + ⋯
L(s)  = 1  + (0.0848 − 0.996i)3-s + (−0.519 + 0.854i)5-s + (−0.506 + 0.862i)7-s + (−0.985 − 0.169i)9-s + (−1.46 + 0.845i)11-s + (0.0984 + 0.0984i)13-s + (0.807 + 0.589i)15-s + (−0.232 + 0.868i)17-s + (−0.552 − 0.319i)19-s + (0.816 + 0.578i)21-s + (−0.221 − 0.825i)23-s + (−0.461 − 0.887i)25-s + (−0.252 + 0.967i)27-s + 1.56·29-s + (−0.131 − 0.227i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.427 - 0.903i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.427 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280602 + 0.443250i\)
\(L(\frac12)\) \(\approx\) \(0.280602 + 0.443250i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.146 + 1.72i)T \)
5 \( 1 + (1.16 - 1.91i)T \)
7 \( 1 + (1.34 - 2.28i)T \)
good11 \( 1 + (4.85 - 2.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.354 - 0.354i)T + 13iT^{2} \)
17 \( 1 + (0.959 - 3.57i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.40 + 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.06 + 3.95i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
31 \( 1 + (0.730 + 1.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.95 - 7.29i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.89iT - 41T^{2} \)
43 \( 1 + (-7.19 - 7.19i)T + 43iT^{2} \)
47 \( 1 + (10.7 - 2.88i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.37 + 0.636i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.64 + 9.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.67 + 2.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.43 - 0.921i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + (2.79 - 10.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.42 - 0.822i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.481 - 0.481i)T - 83iT^{2} \)
89 \( 1 + (7.37 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.38 - 5.38i)T - 97iT^{2} \)
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

−11.55055527622654941871955952671, −10.69787309892437750950052276354, −9.762196339773060378250288143747, −8.345281079176301166974692088875, −7.931106396730333562282065783205, −6.68136907752902580003625969989, −6.20770365778066051012146087569, −4.74477731008242931348222358918, −3.00608774072179806074715592097, −2.27837949809164393165453439925, 0.31080360122054786500310029132, 2.94899546423146710124280333675, 3.99038221313529521606020104000, 4.94492368917215604150676424498, 5.85146592416641133829349392112, 7.42824424599147569405342916910, 8.282029492991980321865781603473, 9.090102238034470061980701783396, 10.11952275561925140148761546232, 10.74819617144284873152771815592

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