Properties

Label 2-420-7.4-c1-0-5
Degree $2$
Conductor $420$
Sign $-0.0633 + 0.997i$
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 + (−1.32 − 2.29i)7-s + (−0.499 − 0.866i)9-s + (0.822 − 1.42i)11-s − 2.64·13-s + 0.999·15-s + (0.822 − 1.42i)17-s + (−4.14 − 7.18i)19-s + 2.64·21-s + (−0.822 − 1.42i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s + 7.64·29-s + (2.14 − 3.71i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.499 − 0.866i)7-s + (−0.166 − 0.288i)9-s + (0.248 − 0.429i)11-s − 0.733·13-s + 0.258·15-s + (0.199 − 0.345i)17-s + (−0.951 − 1.64i)19-s + 0.577·21-s + (−0.171 − 0.297i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s + 1.41·29-s + (0.385 − 0.667i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555043 - 0.591382i\)
\(L(\frac12)\) \(\approx\) \(0.555043 - 0.591382i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 2.29i)T \)
good11 \( 1 + (-0.822 + 1.42i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 + (-0.822 + 1.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.14 + 7.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.822 + 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.64T + 29T^{2} \)
31 \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.322 + 0.559i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 + 5.93T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.64 - 2.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.46 - 9.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.322 - 0.559i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (6.61 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.14 + 1.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + (-7.11 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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

−10.91345419405648271009729866043, −10.08726987741506826181576256842, −9.269224367286378008446015164527, −8.312329249471234876190076533129, −7.11615356215034793730304401592, −6.30017843459664662762383890961, −4.90103082890823506593209326670, −4.19528950172189039769452603207, −2.85068171649318596338223853751, −0.52885149077074766655734548929, 1.93556318074068897418215183173, 3.23440761245805510457951021020, 4.69085356341645749707963176647, 5.98212531102319681356605782646, 6.59823363616323770365279907284, 7.74755754013065176432037022718, 8.561384353522931026361829436276, 9.782435076819715593067800619018, 10.44933674963216971402297660901, 11.67252884373997739638875708613

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