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Properties

Label 2-630-63.58-c1-0-14
Degree $2$
Conductor $630$
Sign $0.965 - 0.259i$
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  − 2-s + (1.72 − 0.132i)3-s + 4-s + (0.5 + 0.866i)5-s + (−1.72 + 0.132i)6-s + (1.01 + 2.44i)7-s − 8-s + (2.96 − 0.457i)9-s + (−0.5 − 0.866i)10-s + (2.91 − 5.04i)11-s + (1.72 − 0.132i)12-s + (−0.511 + 0.885i)13-s + (−1.01 − 2.44i)14-s + (0.978 + 1.42i)15-s + 16-s + (−2.17 − 3.76i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.997 − 0.0765i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.705 + 0.0541i)6-s + (0.384 + 0.922i)7-s − 0.353·8-s + (0.988 − 0.152i)9-s + (−0.158 − 0.273i)10-s + (0.878 − 1.52i)11-s + (0.498 − 0.0382i)12-s + (−0.141 + 0.245i)13-s + (−0.272 − 0.652i)14-s + (0.252 + 0.369i)15-s + 0.250·16-s + (−0.527 − 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.965 - 0.259i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70816 + 0.225683i\)
\(L(\frac12)\) \(\approx\) \(1.70816 + 0.225683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.72 + 0.132i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.01 - 2.44i)T \)
good11 \( 1 + (-2.91 + 5.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.511 - 0.885i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.17 + 3.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.704 - 1.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.30 - 5.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.77 - 4.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.82T + 31T^{2} \)
37 \( 1 + (3.85 - 6.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.46 + 9.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.340 + 0.589i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.58T + 47T^{2} \)
53 \( 1 + (3.17 + 5.50i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.56T + 59T^{2} \)
61 \( 1 - 1.43T + 61T^{2} \)
67 \( 1 - 13.9T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + (0.343 + 0.594i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.35T + 79T^{2} \)
83 \( 1 + (6.94 + 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.81 - 3.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.46 - 9.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.56128501515321394440420728424, −9.244575111156982345943206927341, −9.087871064271746083799882252822, −8.257385273852813664731477362714, −7.23915546071399730210813900309, −6.40521977412854611034260686932, −5.25597759715774361568997138837, −3.59482388319237620639024635302, −2.71377368757915403540708406017, −1.49658977389916115675831477868, 1.35100470718697936698890405560, 2.38950179677767182420383218145, 4.02641032868444369414371388517, 4.65660505554701102992962194139, 6.48846508352888566684356053021, 7.23880610027081593917057718862, 8.023062639837935280019040864080, 8.891440796994629031556018022847, 9.584402618280079122452651278863, 10.31586612485463763467369908637

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