Properties

Label 2-630-63.58-c1-0-11
Degree $2$
Conductor $630$
Sign $0.945 - 0.324i$
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 + (−1.02 + 1.39i)3-s + 4-s + (0.5 + 0.866i)5-s + (1.02 − 1.39i)6-s + (−2.52 + 0.795i)7-s − 8-s + (−0.918 − 2.85i)9-s + (−0.5 − 0.866i)10-s + (2.96 − 5.14i)11-s + (−1.02 + 1.39i)12-s + (2.58 − 4.47i)13-s + (2.52 − 0.795i)14-s + (−1.72 − 0.183i)15-s + 16-s + (3.72 + 6.46i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.588 + 0.808i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (0.416 − 0.571i)6-s + (−0.953 + 0.300i)7-s − 0.353·8-s + (−0.306 − 0.951i)9-s + (−0.158 − 0.273i)10-s + (0.894 − 1.54i)11-s + (−0.294 + 0.404i)12-s + (0.717 − 1.24i)13-s + (0.674 − 0.212i)14-s + (−0.444 − 0.0473i)15-s + 0.250·16-s + (0.904 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.945 - 0.324i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.945 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.847439 + 0.141512i\)
\(L(\frac12)\) \(\approx\) \(0.847439 + 0.141512i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 - 1.39i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.52 - 0.795i)T \)
good11 \( 1 + (-2.96 + 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.58 + 4.47i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.72 - 6.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.78 + 3.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.367 + 0.635i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.46 - 4.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.93T + 31T^{2} \)
37 \( 1 + (-0.0338 + 0.0585i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.03 - 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.84 + 6.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 + (-2.41 - 4.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.55T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 6.07T + 67T^{2} \)
71 \( 1 - 5.08T + 71T^{2} \)
73 \( 1 + (-0.250 - 0.433i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.73T + 79T^{2} \)
83 \( 1 + (2.63 + 4.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.43 + 4.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.484 + 0.839i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.53770899882903709329317453315, −9.945267235189693428217614050224, −8.900348728438647732795867531882, −8.432425692866661983710112189112, −6.89968503463925803058855797134, −5.95159119787599500890298598759, −5.67826156047646465413662526087, −3.62195477544153621109262493417, −3.18495903759112065820338445337, −0.859189649824870529058602089556, 1.04524203634278503185634118968, 2.19154741720087814963968006267, 3.91508962130826350158639148940, 5.26999444067432165756900713395, 6.41112919882615429256600911778, 6.98914356804995285233229347878, 7.68189225065363916949281361868, 9.033359393868298629387271452564, 9.622029946905517233135596406710, 10.34201259818894197623861034090

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