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Properties

Label 2-630-63.5-c1-0-10
Degree $2$
Conductor $630$
Sign $0.571 - 0.820i$
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  + i·2-s + (−1.04 + 1.37i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−1.37 − 1.04i)6-s + (−0.323 − 2.62i)7-s − i·8-s + (−0.797 − 2.89i)9-s + (0.866 − 0.5i)10-s + (0.664 + 0.383i)11-s + (1.04 − 1.37i)12-s + (3.78 + 2.18i)13-s + (2.62 − 0.323i)14-s + (1.71 + 0.219i)15-s + 16-s + (1.15 + 1.99i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.605 + 0.795i)3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.562 − 0.428i)6-s + (−0.122 − 0.992i)7-s − 0.353i·8-s + (−0.265 − 0.964i)9-s + (0.273 − 0.158i)10-s + (0.200 + 0.115i)11-s + (0.302 − 0.397i)12-s + (1.04 + 0.605i)13-s + (0.701 − 0.0863i)14-s + (0.443 + 0.0567i)15-s + 0.250·16-s + (0.279 + 0.483i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.571 - 0.820i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.571 - 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994285 + 0.519459i\)
\(L(\frac12)\) \(\approx\) \(0.994285 + 0.519459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.04 - 1.37i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.323 + 2.62i)T \)
good11 \( 1 + (-0.664 - 0.383i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.78 - 2.18i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.15 - 1.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.19 - 2.42i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.84 + 2.79i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (9.12 - 5.26i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.05iT - 31T^{2} \)
37 \( 1 + (-4.15 + 7.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.65 + 4.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.94 - 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.46T + 47T^{2} \)
53 \( 1 + (6.48 - 3.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 + 9.49iT - 61T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 - 7.50iT - 71T^{2} \)
73 \( 1 + (-10.1 + 5.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.11T + 79T^{2} \)
83 \( 1 + (-0.254 - 0.440i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.80 - 4.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.61 + 4.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.91164637483426186297841410780, −9.593499692020335103032812136430, −9.218627212641098291146228325059, −8.015866884588962933677574773217, −7.11374693194307456731598312558, −6.13465789021202263790394867385, −5.32064940430433531358987345694, −4.13505783154618150325029826367, −3.70662380590805411265125529009, −0.948870600786614929254303292190, 1.04072692306373586860921241749, 2.51217220302869597143010566031, 3.48794528827336368894700937092, 5.16423826154234233434498601282, 5.77494830059742956420277760201, 6.88397408043288029315202668497, 7.84233683635590046866245504408, 8.777826127846650717957061329388, 9.656106386295010835963985335672, 10.82838235722040510742591368819

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