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Properties

Label 2-630-9.4-c1-0-8
Degree $2$
Conductor $630$
Sign $-0.939 - 0.342i$
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 + (0.866 + 1.5i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.866 + 1.5i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (−1.5 + 2.59i)9-s − 0.999·10-s + (−0.133 − 0.232i)11-s − 1.73·12-s + (−0.633 + 1.09i)13-s + (−0.499 + 0.866i)14-s − 1.73·15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 + 0.866i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.353 + 0.612i)6-s + (0.188 + 0.327i)7-s − 0.353·8-s + (−0.5 + 0.866i)9-s − 0.316·10-s + (−0.0403 − 0.0699i)11-s − 0.499·12-s + (−0.175 + 0.304i)13-s + (−0.133 + 0.231i)14-s − 0.447·15-s + (−0.125 − 0.216i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.939 - 0.342i$
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: \((2,\ 630,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307489 + 1.74385i\)
\(L(\frac12)\) \(\approx\) \(0.307489 + 1.74385i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (0.133 + 0.232i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.633 - 1.09i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 0.464T + 19T^{2} \)
23 \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.09 + 1.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.46 + 4.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.66T + 37T^{2} \)
41 \( 1 + (1.13 - 1.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.69 - 8.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.196T + 53T^{2} \)
59 \( 1 + (0.767 - 1.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.36 + 2.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.26T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.73 - 4.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.46T + 89T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.06837710237314808767421725510, −9.827923851146029346162987700483, −9.340166615133044135240767474820, −8.138373344499818939547441229376, −7.71156740581577991286524286256, −6.39627939931430624787537519146, −5.45047715256780961640005466983, −4.46549793846525118456967138717, −3.57136874626710641090714408047, −2.44303820372202626088329681013, 0.833899722570312568576279629044, 2.17621906146300713724279255854, 3.33108164752102398060963313057, 4.42108404038092180675018442219, 5.58340666723193887135008810049, 6.66352577672453161674092754430, 7.66838356007107277751418567301, 8.432017270495681233007873559963, 9.315183015384187670897329149659, 10.29100948798725807460376725035

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