↑

Properties

Label 2-630-315.194-c1-0-42
Degree $2$
Conductor $630$
Sign $-0.845 - 0.533i$
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  − 2-s + (−0.630 − 1.61i)3-s + 4-s + (1.90 + 1.17i)5-s + (0.630 + 1.61i)6-s + (−1.83 − 1.90i)7-s − 8-s + (−2.20 + 2.03i)9-s + (−1.90 − 1.17i)10-s + (−1.13 − 0.652i)11-s + (−0.630 − 1.61i)12-s + (−3.03 + 5.25i)13-s + (1.83 + 1.90i)14-s + (0.687 − 3.81i)15-s + 16-s + (−0.502 + 0.290i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.363 − 0.931i)3-s + 0.5·4-s + (0.852 + 0.523i)5-s + (0.257 + 0.658i)6-s + (−0.693 − 0.720i)7-s − 0.353·8-s + (−0.735 + 0.677i)9-s + (−0.602 − 0.370i)10-s + (−0.340 − 0.196i)11-s + (−0.181 − 0.465i)12-s + (−0.841 + 1.45i)13-s + (0.490 + 0.509i)14-s + (0.177 − 0.984i)15-s + 0.250·16-s + (−0.121 + 0.0704i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.845 - 0.533i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00999319 + 0.0345795i\)
\(L(\frac12)\) \(\approx\) \(0.00999319 + 0.0345795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.630 + 1.61i)T \)
5 \( 1 + (-1.90 - 1.17i)T \)
7 \( 1 + (1.83 + 1.90i)T \)
good11 \( 1 + (1.13 + 0.652i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.03 - 5.25i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.502 - 0.290i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.86 + 3.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.28 + 5.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.125 - 0.0724i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.38iT - 31T^{2} \)
37 \( 1 + (2.02 + 1.17i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.16 + 5.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.0 - 5.81i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.41iT - 47T^{2} \)
53 \( 1 + (4.19 + 7.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.08T + 59T^{2} \)
61 \( 1 - 2.12iT - 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (3.84 + 6.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.09T + 79T^{2} \)
83 \( 1 + (-4.32 + 2.49i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.35 + 2.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.76 - 9.97i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.19297146051421874770282062415, −9.238805760595342602967150030112, −8.349942637614191889740549536812, −7.07442862467294198025226173659, −6.75168318393063335637677348916, −6.03121674968459884664195349733, −4.54720654823837088082050127086, −2.72375788693263740553062174587, −1.87237051856777711286609379860, −0.02318979519295111935112881285, 2.16923117117531585657801373063, 3.36000748846136310996738165209, 4.91825953259004053426700957110, 5.73732355341283419077490088704, 6.35811759761569332788239896568, 7.905900676715550585282973541260, 8.725826993387691838022703884669, 9.581910056302877307837052854442, 10.09686439202380095318623558859, 10.62992325957655842740590158554

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