↑

Properties

Label 2-630-35.3-c1-0-14
Degree $2$
Conductor $630$
Sign $0.871 - 0.489i$
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  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (2.03 − 0.935i)5-s + (1.83 + 1.90i)7-s + (0.707 + 0.707i)8-s + (2.20 − 0.378i)10-s + (−2.01 + 3.49i)11-s + (0.204 − 0.204i)13-s + (1.28 + 2.31i)14-s + (0.500 + 0.866i)16-s + (1.97 − 0.527i)17-s + (−3.10 − 5.37i)19-s + (2.22 + 0.204i)20-s + (−2.85 + 2.85i)22-s + (−1.17 + 4.38i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (0.908 − 0.418i)5-s + (0.695 + 0.718i)7-s + (0.249 + 0.249i)8-s + (0.696 − 0.119i)10-s + (−0.609 + 1.05i)11-s + (0.0568 − 0.0568i)13-s + (0.343 + 0.618i)14-s + (0.125 + 0.216i)16-s + (0.477 − 0.128i)17-s + (−0.711 − 1.23i)19-s + (0.497 + 0.0458i)20-s + (−0.609 + 0.609i)22-s + (−0.244 + 0.914i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.871 - 0.489i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.871 - 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57329 + 0.673627i\)
\(L(\frac12)\) \(\approx\) \(2.57329 + 0.673627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-2.03 + 0.935i)T \)
7 \( 1 + (-1.83 - 1.90i)T \)
good11 \( 1 + (2.01 - 3.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.204 + 0.204i)T - 13iT^{2} \)
17 \( 1 + (-1.97 + 0.527i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.10 + 5.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.17 - 4.38i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.15iT - 29T^{2} \)
31 \( 1 + (-6.33 - 3.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.46 + 1.19i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.58iT - 41T^{2} \)
43 \( 1 + (4.97 + 4.97i)T + 43iT^{2} \)
47 \( 1 + (-0.0815 + 0.304i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.00 + 2.14i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.427 - 0.740i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.99 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.817 + 3.05i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.12T + 71T^{2} \)
73 \( 1 + (2.98 + 11.1i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.39 - 2.53i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.85 + 3.85i)T - 83iT^{2} \)
89 \( 1 + (-1.53 - 2.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.63 + 6.63i)T + 97iT^{2} \)
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

−10.64992979739689918733829308148, −9.843124369883211298937430775949, −8.888670189690616810858839923793, −8.010507790048540094445349336853, −6.97718159870183883355658819168, −5.92175063082341875268726575880, −5.12559588854636907673191007844, −4.48307446583801816015667975374, −2.72640420678823942159950493606, −1.82530149626850066439324096428, 1.44983713360727171187961998959, 2.75841717720257866238299995902, 3.87487998092976172343554563194, 5.06124583448163175063927085368, 5.90696595875971858544806683611, 6.70856548338773192926939311780, 7.86359766017389306875571648512, 8.682333235615689702467299108821, 10.20118298080562575460837481288, 10.43401485595452280474764193063

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