↑

Properties

Label 2-420-105.23-c1-0-13
Degree $2$
Conductor $420$
Sign $0.0882 + 0.996i$
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  + (0.735 − 1.56i)3-s + (1.16 − 1.91i)5-s + (−1.34 + 2.28i)7-s + (−1.91 − 2.30i)9-s + (4.85 − 2.80i)11-s + (0.354 + 0.354i)13-s + (−2.14 − 3.22i)15-s + (0.959 − 3.57i)17-s + (−2.40 − 1.39i)19-s + (2.59 + 3.78i)21-s + (1.06 + 3.95i)23-s + (−2.30 − 4.43i)25-s + (−5.02 + 1.30i)27-s − 8.44·29-s + (−0.730 − 1.26i)31-s + ⋯
L(s)  = 1  + (0.424 − 0.905i)3-s + (0.519 − 0.854i)5-s + (−0.506 + 0.862i)7-s + (−0.639 − 0.769i)9-s + (1.46 − 0.845i)11-s + (0.0984 + 0.0984i)13-s + (−0.553 − 0.832i)15-s + (0.232 − 0.868i)17-s + (−0.552 − 0.319i)19-s + (0.565 + 0.824i)21-s + (0.221 + 0.825i)23-s + (−0.461 − 0.887i)25-s + (−0.967 + 0.252i)27-s − 1.56·29-s + (−0.131 − 0.227i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0882 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0882 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0882 + 0.996i$
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: \((2,\ 420,\ (\ :1/2),\ 0.0882 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21757 - 1.11451i\)
\(L(\frac12)\) \(\approx\) \(1.21757 - 1.11451i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.735 + 1.56i)T \)
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

−11.41384253261785515800149452900, −9.582828263318516655071528879141, −9.096660338178314255979096106095, −8.496327121640355234819232435755, −7.20882404441243050803816235518, −6.17436430361681421179204867459, −5.54596188090459197397371755528, −3.84056210879973642003278447392, −2.50261116708212909137379562177, −1.12526966132951975970317473542, 2.09388563056113217982128000948, 3.64118060276076470928875129098, 4.15353640835893417972311863665, 5.77635876436993061205003459408, 6.72937683165225880138171389680, 7.64000611032991524735028938744, 9.055198469652455328436119764036, 9.603874942275603862343414123727, 10.56410364418386153744939634892, 10.90926676125596341899195122379

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