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Properties

Label 2-630-63.16-c1-0-2
Degree $2$
Conductor $630$
Sign $-0.839 - 0.543i$
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 + (0.702 + 1.58i)3-s + (−0.499 + 0.866i)4-s + 5-s + (1.01 − 1.39i)6-s + (−2.56 − 0.658i)7-s + 0.999·8-s + (−2.01 + 2.22i)9-s + (−0.5 − 0.866i)10-s − 6.44·11-s + (−1.72 − 0.183i)12-s + (0.332 + 0.575i)13-s + (0.710 + 2.54i)14-s + (0.702 + 1.58i)15-s + (−0.5 − 0.866i)16-s + (0.411 + 0.713i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.405 + 0.914i)3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (0.416 − 0.571i)6-s + (−0.968 − 0.249i)7-s + 0.353·8-s + (−0.670 + 0.741i)9-s + (−0.158 − 0.273i)10-s − 1.94·11-s + (−0.497 − 0.0528i)12-s + (0.0920 + 0.159i)13-s + (0.189 + 0.681i)14-s + (0.181 + 0.408i)15-s + (−0.125 − 0.216i)16-s + (0.0998 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123032 + 0.416224i\)
\(L(\frac12)\) \(\approx\) \(0.123032 + 0.416224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.702 - 1.58i)T \)
5 \( 1 - T \)
7 \( 1 + (2.56 + 0.658i)T \)
good11 \( 1 + 6.44T + 11T^{2} \)
13 \( 1 + (-0.332 - 0.575i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.411 - 0.713i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.77 - 4.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.48T + 23T^{2} \)
29 \( 1 + (-2.03 + 3.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.81 + 6.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.32 - 9.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.511 - 0.886i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.15 - 7.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.30 + 2.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.27 + 5.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.45 + 2.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.383 - 0.664i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.64 - 6.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.44T + 71T^{2} \)
73 \( 1 + (-4.71 - 8.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.70 - 4.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.897 + 1.55i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.06 - 3.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.98 - 15.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.49885713223243134945627370976, −10.06664668037781758901037618913, −9.665518293149074559695546489220, −8.325130327131399868089046676529, −7.952834283128333528930207084552, −6.36466424210457047187865069305, −5.33143355938084922346098283632, −4.20886773658088947526013112235, −3.14395124164481923443405951493, −2.27522055743208349530425695811, 0.22888688944300187108818542133, 2.21694483529210823960069661927, 3.15260748959110185665882821794, 5.06435722564729813411055883046, 5.91891844008547399405538196093, 6.78841224989512433951597535058, 7.51606759046608451753149162633, 8.483664471663689166475900796074, 9.091984544680371729014878943291, 10.16382806853957870056867574461

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