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Properties

Label 2-630-7.3-c2-0-20
Degree $2$
Conductor $630$
Sign $0.999 + 0.0230i$
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 + (6.51 − 2.55i)7-s − 2.82·8-s + (−2.73 − 1.58i)10-s + (5.79 − 10.0i)11-s − 7.86i·13-s + (7.73 + 6.17i)14-s + (−2.00 − 3.46i)16-s + (−23.9 − 13.8i)17-s + (27.2 − 15.7i)19-s − 4.47i·20-s + 16.3·22-s + (9.07 + 15.7i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.930 − 0.365i)7-s − 0.353·8-s + (−0.273 − 0.158i)10-s + (0.526 − 0.912i)11-s − 0.604i·13-s + (0.552 + 0.440i)14-s + (−0.125 − 0.216i)16-s + (−1.40 − 0.811i)17-s + (1.43 − 0.826i)19-s − 0.223i·20-s + 0.744·22-s + (0.394 + 0.683i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0230i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0230i$
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: \((2,\ 630,\ (\ :1),\ 0.999 + 0.0230i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.110109846\)
\(L(\frac12)\) \(\approx\) \(2.110109846\)
\(L(2)\) 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 \( 1 + (-6.51 + 2.55i)T \)
good11 \( 1 + (-5.79 + 10.0i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 7.86iT - 169T^{2} \)
17 \( 1 + (23.9 + 13.8i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-27.2 + 15.7i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-9.07 - 15.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 2.30T + 841T^{2} \)
31 \( 1 + (4.55 + 2.63i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (0.993 + 1.72i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 22.1iT - 1.68e3T^{2} \)
43 \( 1 - 49.8T + 1.84e3T^{2} \)
47 \( 1 + (-66.3 + 38.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-28.5 + 49.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-60.9 - 35.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (58.5 - 33.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (49.0 - 85.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 34.2T + 5.04e3T^{2} \)
73 \( 1 + (-16.8 - 9.74i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (45.2 + 78.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 133. iT - 6.88e3T^{2} \)
89 \( 1 + (-9.58 + 5.53i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 72.3iT - 9.40e3T^{2} \)
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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

−10.57350219690242090528709572108, −9.209580393728044398182779439725, −8.584105408162495030583389381383, −7.49042201738187696546345324141, −7.04820364090728638931309866121, −5.76352036925488288054022309204, −4.91018838486160252479390456944, −3.91973624536493682119435379438, −2.77768009862058358132874165466, −0.791910833649271225249856522932, 1.34380699926776485565407898054, 2.37519188733602670015242962638, 3.98101722927368960907893966862, 4.57251208447601814981419624873, 5.62035600747371109730275381279, 6.81269822316484868835700866341, 7.82648013129138579204844313262, 8.854843899524788222576244745545, 9.458712546317935577579431964130, 10.68272756080561182152478920401

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