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Properties

Label 2-630-9.4-c1-0-6
Degree $2$
Conductor $630$
Sign $0.711 - 0.702i$
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 + (−0.805 + 1.53i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (1.73 − 0.0696i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−1.70 − 2.46i)9-s − 0.999·10-s + (1.12 + 1.94i)11-s + (−0.925 − 1.46i)12-s + (0.334 − 0.578i)13-s + (0.499 − 0.866i)14-s + (0.925 + 1.46i)15-s + (−0.5 − 0.866i)16-s + 6.64·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.464 + 0.885i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.706 − 0.0284i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.567 − 0.823i)9-s − 0.316·10-s + (0.337 + 0.585i)11-s + (−0.267 − 0.422i)12-s + (0.0926 − 0.160i)13-s + (0.133 − 0.231i)14-s + (0.238 + 0.377i)15-s + (−0.125 − 0.216i)16-s + 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.711 - 0.702i$
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: \((2,\ 630,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.956458 + 0.392333i\)
\(L(\frac12)\) \(\approx\) \(0.956458 + 0.392333i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.805 - 1.53i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good11 \( 1 + (-1.12 - 1.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.334 + 0.578i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.64T + 17T^{2} \)
19 \( 1 + 3.40T + 19T^{2} \)
23 \( 1 + (1.62 - 2.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.69 - 6.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.40 - 7.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.07T + 37T^{2} \)
41 \( 1 + (-0.165 + 0.287i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.45 - 2.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.69 - 6.39i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.73T + 53T^{2} \)
59 \( 1 + (3.94 - 6.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.44 - 5.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.27 + 2.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 2.33T + 73T^{2} \)
79 \( 1 + (3.95 + 6.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.62 + 6.27i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + (8.05 + 13.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.52449814163337221176061607461, −9.982810157080743479952728728592, −9.131970065976778700031920808657, −8.479678185367305142543002782645, −7.26436789510923565486262893575, −5.91603142110412242923704000452, −5.09603989032143119579788623524, −4.10523337064175282566234090651, −3.02030198324333678972406059860, −1.34668648562098888133853189551, 0.77612964108909334366187999742, 2.27834248342654387553550315406, 3.96843107493488924941234525145, 5.43298938531259412001115612465, 6.11799995752142697586661815088, 6.88875963440415029777623175760, 7.82871079710261697042160156469, 8.382119741597786726659883799851, 9.641762749163660755859966956319, 10.48776143369174704336764378312

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