Properties

Label 2-630-9.4-c1-0-2
Degree $2$
Conductor $630$
Sign $-0.635 + 0.771i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.724 + 1.57i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1.72 + 0.158i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−1.94 − 2.28i)9-s − 0.999·10-s + (0.275 + 0.476i)11-s + (−1 − 1.41i)12-s + (−2.22 + 3.85i)13-s + (0.499 − 0.866i)14-s + (−1 − 1.41i)15-s + (−0.5 − 0.866i)16-s − 3·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.418 + 0.908i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.704 + 0.0648i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.649 − 0.760i)9-s − 0.316·10-s + (0.0829 + 0.143i)11-s + (−0.288 − 0.408i)12-s + (−0.617 + 1.06i)13-s + (0.133 − 0.231i)14-s + (−0.258 − 0.365i)15-s + (−0.125 − 0.216i)16-s − 0.727·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.635 + 0.771i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.635 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.241558 - 0.511843i\)
\(L(\frac12)\) \(\approx\) \(0.241558 - 0.511843i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.724 - 1.57i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good11 \( 1 + (-0.275 - 0.476i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.22 - 3.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 5.89T + 19T^{2} \)
23 \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.55T + 37T^{2} \)
41 \( 1 + (-5.72 + 9.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.94 - 5.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.89 - 8.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.34T + 53T^{2} \)
59 \( 1 + (3.94 - 6.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.67 - 4.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.84 - 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.44T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.55 - 6.14i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + (-7.84 - 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.02855909305763266724105695831, −10.39493867417529589322236414126, −9.299132926260239378151538291047, −8.711790873925217765685638466973, −7.35732363653874391606270715404, −6.61568588896758886937402658648, −5.78555221685826586918175949462, −4.39409358381716813073097864783, −4.18095450002394439835587017002, −2.64026001773433143198087097211, 0.27429283714470285503377670198, 1.89658907327213209975539413863, 3.01028986725345669538049307976, 4.47850422501264974922527697330, 5.45293387744374220769990100715, 6.26362422706255571231920052194, 7.34508524650094070792123615886, 8.318259508424400016053931137350, 9.122916231389387713930640220282, 10.33302794347077239239209962606

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