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Properties

Label 2-630-35.12-c1-0-3
Degree $2$
Conductor $630$
Sign $0.616 - 0.787i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−2.20 + 0.378i)5-s + (0.126 + 2.64i)7-s + (0.707 − 0.707i)8-s + (−2.03 + 0.935i)10-s + (2.81 + 4.87i)11-s + (−1.42 − 1.42i)13-s + (0.806 + 2.51i)14-s + (0.500 − 0.866i)16-s + (5.12 + 1.37i)17-s + (−1.94 + 3.37i)19-s + (−1.71 + 1.42i)20-s + (3.97 + 3.97i)22-s + (0.290 + 1.08i)23-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.985 + 0.169i)5-s + (0.0477 + 0.998i)7-s + (0.249 − 0.249i)8-s + (−0.642 + 0.295i)10-s + (0.848 + 1.46i)11-s + (−0.396 − 0.396i)13-s + (0.215 + 0.673i)14-s + (0.125 − 0.216i)16-s + (1.24 + 0.333i)17-s + (−0.446 + 0.773i)19-s + (−0.384 + 0.319i)20-s + (0.848 + 0.848i)22-s + (0.0606 + 0.226i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.616 - 0.787i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.616 - 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67633 + 0.816427i\)
\(L(\frac12)\) \(\approx\) \(1.67633 + 0.816427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (2.20 - 0.378i)T \)
7 \( 1 + (-0.126 - 2.64i)T \)
good11 \( 1 + (-2.81 - 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.42 + 1.42i)T + 13iT^{2} \)
17 \( 1 + (-5.12 - 1.37i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.94 - 3.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.290 - 1.08i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 3.15iT - 29T^{2} \)
31 \( 1 + (3.33 - 1.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.86 + 1.30i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (-1.85 + 1.85i)T - 43iT^{2} \)
47 \( 1 + (-1.52 - 5.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.33 + 0.357i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.73 + 4.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 + 2.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.218 + 0.816i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 + (-1.45 + 5.42i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.41 - 3.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.67 + 5.67i)T + 83iT^{2} \)
89 \( 1 + (-5.96 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.63 - 6.63i)T - 97iT^{2} \)
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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

−10.87953915729989366218249050237, −9.958446471718874325351239583660, −9.043460504091149440334885187491, −7.88624294353729199618812716280, −7.20394918562390158014693350190, −6.10049592812679091267587042134, −5.09620220200329785196342296140, −4.12290250454635335395170106837, −3.15881057735837057572506694130, −1.79629812043631793362247417639, 0.882854889155919811856378915011, 3.06667558546848547771754561522, 3.92100101848708819390041155330, 4.68990505693547993220869463860, 5.95834567991953290656076991170, 6.92239505728064722626002974204, 7.69844091800040801968990041004, 8.503303115011667397109360701998, 9.595114480135332567624275120302, 10.83162135268034461660641425103

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