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Properties

Label 2-630-7.3-c2-0-15
Degree $2$
Conductor $630$
Sign $0.939 + 0.343i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + (6.99 − 0.325i)7-s + 2.82·8-s + (−2.73 − 1.58i)10-s + (6.09 − 10.5i)11-s + 25.3i·13-s + (−5.34 − 8.33i)14-s + (−2.00 − 3.46i)16-s + (24.9 + 14.3i)17-s + (−13.9 + 8.03i)19-s + 4.47i·20-s − 17.2·22-s + (−11.8 − 20.5i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (0.998 − 0.0465i)7-s + 0.353·8-s + (−0.273 − 0.158i)10-s + (0.554 − 0.959i)11-s + 1.95i·13-s + (−0.381 − 0.595i)14-s + (−0.125 − 0.216i)16-s + (1.46 + 0.846i)17-s + (−0.732 + 0.422i)19-s + 0.223i·20-s − 0.783·22-s + (−0.516 − 0.893i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.939 + 0.343i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ 0.939 + 0.343i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.849619122\)
\(L(\frac12)\) \(\approx\) \(1.849619122\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-6.99 + 0.325i)T \)
good11 \( 1 + (-6.09 + 10.5i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 25.3iT - 169T^{2} \)
17 \( 1 + (-24.9 - 14.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13.9 - 8.03i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (11.8 + 20.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 27.9T + 841T^{2} \)
31 \( 1 + (-20.0 - 11.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.5 - 25.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 56.9iT - 1.68e3T^{2} \)
43 \( 1 - 7.83T + 1.84e3T^{2} \)
47 \( 1 + (-19.7 + 11.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-24.2 + 42.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-62.3 - 36.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-99.2 + 57.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-35.2 + 61.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 6.41T + 5.04e3T^{2} \)
73 \( 1 + (34.2 + 19.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-27.3 - 47.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 135. iT - 6.88e3T^{2} \)
89 \( 1 + (124. - 72.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 78.9iT - 9.40e3T^{2} \)
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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

−10.33565104634431298946661238360, −9.536845573438543755815198384995, −8.543088827852471547660622131993, −8.152742814597433135945623078538, −6.75797253710012532093591955861, −5.81074551121768413626405112898, −4.53181661229607926730447072339, −3.72283185448492681212379261804, −2.08775982436029071383308766363, −1.21477370533730539755552371934, 0.958521681036946488835771468044, 2.39058054848173948202655620452, 3.96914060921806807953831478088, 5.32441087395712589176420191612, 5.69255628554326322567095738530, 7.19431810844784377526497256939, 7.66109032809036901834123118000, 8.561845363641147814653133153360, 9.654666931907653363152723372069, 10.20656152346811633743899592578

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