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Properties

Label 2-630-63.59-c1-0-30
Degree $2$
Conductor $630$
Sign $-0.349 + 0.937i$
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  + (−0.866 − 0.5i)2-s + (1.34 − 1.09i)3-s + (0.499 + 0.866i)4-s + 5-s + (−1.70 + 0.275i)6-s + (1.16 − 2.37i)7-s − 0.999i·8-s + (0.606 − 2.93i)9-s + (−0.866 − 0.5i)10-s − 1.71i·11-s + (1.61 + 0.615i)12-s + (−4.83 − 2.78i)13-s + (−2.19 + 1.47i)14-s + (1.34 − 1.09i)15-s + (−0.5 + 0.866i)16-s + (−2.40 + 4.17i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.775 − 0.631i)3-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.698 + 0.112i)6-s + (0.438 − 0.898i)7-s − 0.353i·8-s + (0.202 − 0.979i)9-s + (−0.273 − 0.158i)10-s − 0.517i·11-s + (0.467 + 0.177i)12-s + (−1.33 − 0.773i)13-s + (−0.586 + 0.395i)14-s + (0.346 − 0.282i)15-s + (−0.125 + 0.216i)16-s + (−0.584 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.349 + 0.937i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.349 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.843937 - 1.21495i\)
\(L(\frac12)\) \(\approx\) \(0.843937 - 1.21495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.34 + 1.09i)T \)
5 \( 1 - T \)
7 \( 1 + (-1.16 + 2.37i)T \)
good11 \( 1 + 1.71iT - 11T^{2} \)
13 \( 1 + (4.83 + 2.78i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.40 - 4.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.89 + 1.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 + (-7.26 + 4.19i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.65 + 3.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.51 + 7.81i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.42 - 5.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.01 - 6.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.64 - 4.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.80 - 2.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.94 + 8.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.9 - 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.35 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.04iT - 71T^{2} \)
73 \( 1 + (2.87 + 1.65i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.63 - 2.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.50 - 9.53i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.463 + 0.803i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.6 + 6.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.10189586608996960377139186201, −9.588389375259125174155778272672, −8.441811273161053914549532927723, −7.84867805148231510123145571216, −7.09822722032695545871052431847, −6.05688437020626382534106982553, −4.53917072970564680263747296310, −3.26680893892546447254188159353, −2.22202289385665153490774204080, −0.921659139697767312087313648936, 2.01623551298401971898555639058, 2.78810971527502943885612318104, 4.71872712498465576060666528300, 5.10218846553925404624490382355, 6.65462510301883346373567297168, 7.40708914860365686612336407738, 8.677327209326845568167374588968, 8.884025370480901047392031226708, 9.978003541367467776077833716449, 10.31702595205707493448038890568

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