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Properties

Label 2-630-9.7-c1-0-15
Degree $2$
Conductor $630$
Sign $-0.173 + 0.984i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + 1.73i·3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−1.49 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 2.99·9-s + 0.999·10-s + (−1.5 + 2.59i)11-s + (1.49 − 0.866i)12-s + (−2.5 − 4.33i)13-s + (−0.499 − 0.866i)14-s + (1.49 − 0.866i)15-s + (−0.5 + 0.866i)16-s − 3·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + 0.999i·3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.612 − 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s − 0.999·9-s + 0.316·10-s + (−0.452 + 0.783i)11-s + (0.433 − 0.249i)12-s + (−0.693 − 1.20i)13-s + (−0.133 − 0.231i)14-s + (0.387 − 0.223i)15-s + (−0.125 + 0.216i)16-s − 0.727·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.25319999840113193605985791104, −9.432220316330549210679677317689, −8.675491434505950154151157744082, −7.87569248634689643462928193979, −6.85974165607232255542363724809, −5.51107136098992982734167394184, −5.04857049395821607469841694228, −3.92150673435308056670677185243, −2.49812221506416199255590816022, 0, 1.75220651030138585254562129753, 2.85542466720270114539085431931, 3.98912654714384482663577577652, 5.46784927278664523162044606806, 6.64805201807973562484538792722, 7.36035092887956838632126613266, 8.169120215378619190584539822270, 9.110278112615298860805625974258, 9.982347663495477407311000425765

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