↑

Properties

Label 2-630-63.47-c1-0-3
Degree $2$
Conductor $630$
Sign $-0.144 - 0.989i$
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  + (−0.866 + 0.5i)2-s + (−1.69 − 0.348i)3-s + (0.499 − 0.866i)4-s + 5-s + (1.64 − 0.546i)6-s + (−1.74 + 1.99i)7-s + 0.999i·8-s + (2.75 + 1.18i)9-s + (−0.866 + 0.5i)10-s − 1.76i·11-s + (−1.15 + 1.29i)12-s + (1.79 − 1.03i)13-s + (0.514 − 2.59i)14-s + (−1.69 − 0.348i)15-s + (−0.5 − 0.866i)16-s + (0.861 + 1.49i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.979 − 0.201i)3-s + (0.249 − 0.433i)4-s + 0.447·5-s + (0.670 − 0.223i)6-s + (−0.658 + 0.752i)7-s + 0.353i·8-s + (0.919 + 0.394i)9-s + (−0.273 + 0.158i)10-s − 0.532i·11-s + (−0.331 + 0.373i)12-s + (0.499 − 0.288i)13-s + (0.137 − 0.693i)14-s + (−0.438 − 0.0899i)15-s + (−0.125 − 0.216i)16-s + (0.208 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399573 + 0.462271i\)
\(L(\frac12)\) \(\approx\) \(0.399573 + 0.462271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (1.69 + 0.348i)T \)
5 \( 1 - T \)
7 \( 1 + (1.74 - 1.99i)T \)
good11 \( 1 + 1.76iT - 11T^{2} \)
13 \( 1 + (-1.79 + 1.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.861 - 1.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.79 + 3.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.15iT - 23T^{2} \)
29 \( 1 + (-5.62 - 3.24i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.23 - 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.24 - 9.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.17 - 5.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.94 + 3.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.01 - 10.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.34 - 2.51i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.25 - 7.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.00 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.12 - 3.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.97iT - 71T^{2} \)
73 \( 1 + (9.43 - 5.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.83 - 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.77 + 10.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.36 + 7.55i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.31 + 3.06i)T + (48.5 + 84.0i)T^{2} \)
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

−10.69303640222549221235882208131, −10.06385619848434633122181803913, −9.053209747394478086789856055374, −8.338482599464019315174994895723, −7.08511886241620346185202018139, −6.21119495597288785455684902722, −5.82440642479299268749347725522, −4.65957312883628446574089677651, −2.89847787008356346482793615843, −1.31804293413117842113646344107, 0.50211126910590205202719703714, 2.09982167963177223983099431520, 3.80174623067177150676668965782, 4.64317205821227830208265281882, 6.18617850188129946488972727916, 6.57678699403713096790573450850, 7.66639500964645355228031407836, 8.862018242946119674410681808184, 9.750634892618738716249846726482, 10.61272234337736710896533237649

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