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Properties

Label 2-630-315.209-c1-0-20
Degree $2$
Conductor $630$
Sign $0.746 - 0.665i$
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  + (−0.5 + 0.866i)2-s + (1.73 − 0.0696i)3-s + (−0.499 − 0.866i)4-s + (2.09 − 0.790i)5-s + (−0.805 + 1.53i)6-s + (−2.36 + 1.17i)7-s + 0.999·8-s + (2.99 − 0.240i)9-s + (−0.360 + 2.20i)10-s + (1.16 + 0.673i)11-s + (−0.925 − 1.46i)12-s + (1.23 + 2.13i)13-s + (0.165 − 2.64i)14-s + (3.56 − 1.51i)15-s + (−0.5 + 0.866i)16-s − 0.246i·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.999 − 0.0401i)3-s + (−0.249 − 0.433i)4-s + (0.935 − 0.353i)5-s + (−0.328 + 0.626i)6-s + (−0.895 + 0.444i)7-s + 0.353·8-s + (0.996 − 0.0803i)9-s + (−0.114 + 0.697i)10-s + (0.351 + 0.202i)11-s + (−0.267 − 0.422i)12-s + (0.341 + 0.591i)13-s + (0.0443 − 0.705i)14-s + (0.920 − 0.390i)15-s + (−0.125 + 0.216i)16-s − 0.0597i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.746 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79913 + 0.685621i\)
\(L(\frac12)\) \(\approx\) \(1.79913 + 0.685621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.73 + 0.0696i)T \)
5 \( 1 + (-2.09 + 0.790i)T \)
7 \( 1 + (2.36 - 1.17i)T \)
good11 \( 1 + (-1.16 - 0.673i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.23 - 2.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.246iT - 17T^{2} \)
19 \( 1 + 1.47iT - 19T^{2} \)
23 \( 1 + (-4.15 - 7.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.56 + 2.63i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.15 + 4.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.87iT - 37T^{2} \)
41 \( 1 + (4.73 + 8.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.27 - 4.20i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.27 + 4.77i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.10T + 53T^{2} \)
59 \( 1 + (-2.66 - 4.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.46 + 3.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.74 - 1.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + 5.92T + 73T^{2} \)
79 \( 1 + (-0.270 + 0.467i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.04 + 0.603i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + (5.12 - 8.88i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.15067792419708685921640792060, −9.438422352861915437314610496651, −9.173405711088032046920095940678, −8.234864093684478714924175509917, −7.10334398672880757401366302682, −6.39553298607184456055932374197, −5.40112304805716191735286709335, −4.12617494131510284050279251415, −2.79361185670011937354955115128, −1.52592429347453209175980180451, 1.33607462362848964089236531312, 2.77734342197317686721398154250, 3.34991529875749108737662256521, 4.64969636166077436867326968411, 6.23177861496839797829881897908, 6.97777309499898947028149427783, 8.124767250133694460655071910932, 8.981618919012482367251797261621, 9.651373543443426907987571878491, 10.36422215141299479019837123529

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