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Properties

Label 2-630-63.59-c1-0-6
Degree $2$
Conductor $630$
Sign $-0.279 - 0.960i$
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  + (0.866 + 0.5i)2-s + (−1.69 − 0.338i)3-s + (0.499 + 0.866i)4-s + 5-s + (−1.30 − 1.14i)6-s + (−2.46 + 0.968i)7-s + 0.999i·8-s + (2.77 + 1.14i)9-s + (0.866 + 0.5i)10-s − 0.656i·11-s + (−0.556 − 1.64i)12-s + (2.39 + 1.38i)13-s + (−2.61 − 0.392i)14-s + (−1.69 − 0.338i)15-s + (−0.5 + 0.866i)16-s + (−3.36 + 5.83i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.980 − 0.195i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.531 − 0.466i)6-s + (−0.930 + 0.366i)7-s + 0.353i·8-s + (0.923 + 0.382i)9-s + (0.273 + 0.158i)10-s − 0.197i·11-s + (−0.160 − 0.473i)12-s + (0.664 + 0.383i)13-s + (−0.699 − 0.104i)14-s + (−0.438 − 0.0873i)15-s + (−0.125 + 0.216i)16-s + (−0.817 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.279 - 0.960i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.279 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.764755 + 1.01891i\)
\(L(\frac12)\) \(\approx\) \(0.764755 + 1.01891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (1.69 + 0.338i)T \)
5 \( 1 - T \)
7 \( 1 + (2.46 - 0.968i)T \)
good11 \( 1 + 0.656iT - 11T^{2} \)
13 \( 1 + (-2.39 - 1.38i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.36 - 5.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.633 - 0.365i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.20iT - 23T^{2} \)
29 \( 1 + (-6.21 + 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.93 - 4.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.05 - 1.81i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.35 - 5.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.29 - 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.92 + 5.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.77 + 3.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.62 + 8.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.35 - 0.782i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.28 - 5.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (-5.95 - 3.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.53 + 13.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.71 + 15.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.32 + 2.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.02 + 2.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.02403763804122325007378681225, −10.11873544397959561010992637818, −9.164723038344995548558915941003, −8.080164612493009630455457192027, −6.83034741437459377116667310768, −6.22729483889617048057015438078, −5.68648558578527632230720974820, −4.49533134882874698583401424956, −3.39619501985551884135412596046, −1.75640649574337628064271438361, 0.64485933328790679446040362447, 2.50455799624188299756295917715, 3.83395407126402897040374374468, 4.78394030906456846190593415292, 5.74100338356442287037740165363, 6.57102619018359634914030971982, 7.18863478751320417000566732644, 8.960398785412115955895941650033, 9.700179066512592266035691885880, 10.72698425831082716423450098068

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