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Properties

Label 2-630-63.59-c1-0-28
Degree $2$
Conductor $630$
Sign $0.964 - 0.263i$
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 + (1.71 − 0.227i)3-s + (0.499 + 0.866i)4-s + 5-s + (1.60 + 0.661i)6-s + (0.778 − 2.52i)7-s + 0.999i·8-s + (2.89 − 0.782i)9-s + (0.866 + 0.5i)10-s + 2.90i·11-s + (1.05 + 1.37i)12-s + (−1.69 − 0.979i)13-s + (1.93 − 1.80i)14-s + (1.71 − 0.227i)15-s + (−0.5 + 0.866i)16-s + (1.68 − 2.91i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.991 − 0.131i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (0.653 + 0.269i)6-s + (0.294 − 0.955i)7-s + 0.353i·8-s + (0.965 − 0.260i)9-s + (0.273 + 0.158i)10-s + 0.876i·11-s + (0.304 + 0.396i)12-s + (−0.470 − 0.271i)13-s + (0.518 − 0.481i)14-s + (0.443 − 0.0588i)15-s + (−0.125 + 0.216i)16-s + (0.408 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.964 - 0.263i$
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: \((2,\ 630,\ (\ :1/2),\ 0.964 - 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.07411 + 0.411901i\)
\(L(\frac12)\) \(\approx\) \(3.07411 + 0.411901i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.227i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.778 + 2.52i)T \)
good11 \( 1 - 2.90iT - 11T^{2} \)
13 \( 1 + (1.69 + 0.979i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.97 - 4.02i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.85iT - 23T^{2} \)
29 \( 1 + (-1.87 + 1.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.90 - 1.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.68 - 2.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.24 - 7.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.991 - 1.71i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.323 - 0.560i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.25 + 5.34i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.55 - 7.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.63 - 1.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.56 - 13.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.94iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.26i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.14 - 3.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.87 + 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.43 + 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.21 - 3.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.27491211061393200549377829571, −9.967320188061771243912550008888, −8.681259177797388455382989802884, −7.85999793706496921430357633822, −7.13757167246942389979506802170, −6.31468623944273937396172021784, −4.81985003561115032651226644917, −4.17378989524934122970275150992, −2.92358010611178500917539275535, −1.74424209783207580916838821394, 1.85423030826364842690116052635, 2.68279633561808704802719489495, 3.80287125127623010646313095909, 4.91707464317114217191709208223, 5.88699717062087819631583925107, 6.89051886583897078845746988418, 8.195371750744406827992828481721, 8.849253894657355046141914525541, 9.622540321166925923554245275052, 10.62026284413538158195380305775

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