Properties

Label 2-630-63.59-c1-0-15
Degree $2$
Conductor $630$
Sign $-0.225 + 0.974i$
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 + (−1.35 − 1.08i)3-s + (0.499 + 0.866i)4-s + 5-s + (0.630 + 1.61i)6-s + (−1.80 + 1.93i)7-s − 0.999i·8-s + (0.657 + 2.92i)9-s + (−0.866 − 0.5i)10-s − 0.781i·11-s + (0.260 − 1.71i)12-s + (−2.26 − 1.31i)13-s + (2.52 − 0.774i)14-s + (−1.35 − 1.08i)15-s + (−0.5 + 0.866i)16-s + (3.02 − 5.24i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.780 − 0.624i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (0.257 + 0.658i)6-s + (−0.681 + 0.731i)7-s − 0.353i·8-s + (0.219 + 0.975i)9-s + (−0.273 − 0.158i)10-s − 0.235i·11-s + (0.0753 − 0.494i)12-s + (−0.629 − 0.363i)13-s + (0.676 − 0.206i)14-s + (−0.349 − 0.279i)15-s + (−0.125 + 0.216i)16-s + (0.734 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.225 + 0.974i$
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: \((2,\ 630,\ (\ :1/2),\ -0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443905 - 0.558599i\)
\(L(\frac12)\) \(\approx\) \(0.443905 - 0.558599i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 1.08i)T \)
5 \( 1 - T \)
7 \( 1 + (1.80 - 1.93i)T \)
good11 \( 1 + 0.781iT - 11T^{2} \)
13 \( 1 + (2.26 + 1.31i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.02 + 5.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.09 + 3.51i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.60iT - 23T^{2} \)
29 \( 1 + (-3.96 + 2.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.47 - 3.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.42 + 9.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.46 + 9.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.45 + 7.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.501 + 0.868i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.30 - 4.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.32 - 2.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.84 - 1.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.02 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (-7.67 - 4.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.96 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.03 + 3.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.78 + 6.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.00 + 2.88i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.30890281046171739913090542457, −9.527031815065787818914698039773, −8.843734877977047637370777149508, −7.37141483379211134278881214556, −7.12185525859168799755286293444, −5.65110493892401722436115137559, −5.27647506720904413868914682674, −3.23987619915442324315656494814, −2.19919959138182678682870104982, −0.59367403926084281956554422918, 1.25517302375833799860909737046, 3.27008639943174213731642903599, 4.48558475133516979140909768278, 5.57396389047789424783290525329, 6.38344214942938316532503806033, 7.15175614597198676048100916099, 8.246180387867273079430401626017, 9.481772215583059693150424274580, 10.00783366196553014525936359760, 10.42032670056685733788621279448

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