↑

Properties

Label 2-630-7.4-c1-0-4
Degree $2$
Conductor $630$
Sign $0.895 - 0.444i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
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)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 2.64·7-s + 0.999·8-s + (0.499 − 0.866i)10-s + (−2.32 + 4.02i)11-s + 0.645·13-s + (−1.32 − 2.29i)14-s + (−0.5 − 0.866i)16-s + (−1.64 + 2.85i)17-s + (2.14 + 3.71i)19-s − 0.999·20-s + 4.64·22-s + (−1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 0.999·7-s + 0.353·8-s + (0.158 − 0.273i)10-s + (−0.700 + 1.21i)11-s + 0.179·13-s + (−0.353 − 0.612i)14-s + (−0.125 − 0.216i)16-s + (−0.399 + 0.691i)17-s + (0.492 + 0.852i)19-s − 0.223·20-s + 0.990·22-s + (−0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23835 + 0.290098i\)
\(L(\frac12)\) \(\approx\) \(1.23835 + 0.290098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - 2.64T \)
good11 \( 1 + (2.32 - 4.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.645T + 13T^{2} \)
17 \( 1 + (1.64 - 2.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.14 - 3.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.96 - 5.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.35T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (-4.79 - 8.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.145 + 0.252i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.64 + 8.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.64 + 9.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.35 - 4.07i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.70T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.64 + 9.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 + (3.29 + 5.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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

−10.60183012282064000814150008481, −10.01209613137371274514016751969, −9.024064911183213213532145967615, −7.986602937032356472424316580023, −7.46135177836099986576030576010, −6.15100513759877047079512601372, −4.94904562287620993098124760628, −4.05050023156599942749922471571, −2.57347747478438572284805564125, −1.59975118205444286787532785935, 0.844968877997970793133417862600, 2.51101130923486219526674902761, 4.18626234019313579325708576370, 5.30212313205767893440730274303, 5.81795584716975753447124238639, 7.19831218208569147526507128541, 7.902778210977674892620048638419, 8.776575661882059416559475986395, 9.346937456676844014379775633163, 10.62075260630719574155815492332

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