↑

Properties

Label 2-630-315.299-c1-0-10
Degree $2$
Conductor $630$
Sign $-0.996 - 0.0868i$
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 + (−0.558 + 1.63i)3-s + (−0.499 + 0.866i)4-s + (2.07 + 0.824i)5-s + (−1.69 + 0.335i)6-s + (−0.837 + 2.50i)7-s − 0.999·8-s + (−2.37 − 1.83i)9-s + (0.325 + 2.21i)10-s + 0.811i·11-s + (−1.14 − 1.30i)12-s + (0.0126 + 0.0219i)13-s + (−2.59 + 0.529i)14-s + (−2.51 + 2.94i)15-s + (−0.5 − 0.866i)16-s + (−2.78 + 1.60i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.322 + 0.946i)3-s + (−0.249 + 0.433i)4-s + (0.929 + 0.368i)5-s + (−0.693 + 0.137i)6-s + (−0.316 + 0.948i)7-s − 0.353·8-s + (−0.791 − 0.610i)9-s + (0.102 + 0.699i)10-s + 0.244i·11-s + (−0.329 − 0.376i)12-s + (0.00351 + 0.00609i)13-s + (−0.692 + 0.141i)14-s + (−0.648 + 0.761i)15-s + (−0.125 − 0.216i)16-s + (−0.675 + 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.996 - 0.0868i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.996 - 0.0868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0656191 + 1.50838i\)
\(L(\frac12)\) \(\approx\) \(0.0656191 + 1.50838i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.558 - 1.63i)T \)
5 \( 1 + (-2.07 - 0.824i)T \)
7 \( 1 + (0.837 - 2.50i)T \)
good11 \( 1 - 0.811iT - 11T^{2} \)
13 \( 1 + (-0.0126 - 0.0219i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.78 - 1.60i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.22 - 0.705i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 + (4.38 + 2.52i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.89 - 1.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.76 + 5.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.973 - 1.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.76 - 1.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.50 + 3.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.33 - 9.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.80 - 3.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.527 + 0.304i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.879 + 0.507i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (6.25 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.845 + 1.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.5 - 8.40i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.31 + 4.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.19 + 5.52i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.87160791070411886589002726318, −10.09285101250282856792152342906, −9.135386218324538463049475461718, −8.784165516616388958725858025215, −7.23362415469927314335378988599, −6.20888885287809799366026822942, −5.65601189018974798353954512856, −4.80646154470393125935089837747, −3.54379473808671659338587298029, −2.40690108102260999885200093569, 0.76572977419512265808739006908, 1.96857098099315898309975740789, 3.20105587660517925494599133791, 4.69430673835123213714711828458, 5.56642902991804613382149393331, 6.55321365218853816380447802947, 7.25829904522093751635749734545, 8.565772061544747513730989788031, 9.397043789768772187494457502560, 10.41679750423608037010501405960

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