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Properties

Label 2-630-315.299-c1-0-4
Degree $2$
Conductor $630$
Sign $-0.884 - 0.465i$
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 + (−1.71 − 0.256i)3-s + (−0.499 + 0.866i)4-s + (0.390 + 2.20i)5-s + (−0.633 − 1.61i)6-s + (0.160 − 2.64i)7-s − 0.999·8-s + (2.86 + 0.880i)9-s + (−1.71 + 1.43i)10-s + 1.17i·11-s + (1.07 − 1.35i)12-s + (2.14 + 3.71i)13-s + (2.36 − 1.18i)14-s + (−0.102 − 3.87i)15-s + (−0.5 − 0.866i)16-s + (−6.22 + 3.59i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.988 − 0.148i)3-s + (−0.249 + 0.433i)4-s + (0.174 + 0.984i)5-s + (−0.258 − 0.658i)6-s + (0.0606 − 0.998i)7-s − 0.353·8-s + (0.955 + 0.293i)9-s + (−0.541 + 0.454i)10-s + 0.353i·11-s + (0.311 − 0.391i)12-s + (0.594 + 1.02i)13-s + (0.632 − 0.315i)14-s + (−0.0264 − 0.999i)15-s + (−0.125 − 0.216i)16-s + (−1.50 + 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.884 - 0.465i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.884 - 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228267 + 0.923888i\)
\(L(\frac12)\) \(\approx\) \(0.228267 + 0.923888i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.256i)T \)
5 \( 1 + (-0.390 - 2.20i)T \)
7 \( 1 + (-0.160 + 2.64i)T \)
good11 \( 1 - 1.17iT - 11T^{2} \)
13 \( 1 + (-2.14 - 3.71i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.22 - 3.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.60 - 3.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.91T + 23T^{2} \)
29 \( 1 + (7.50 + 4.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.82 + 1.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.428 + 0.247i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.59 - 9.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.14 - 2.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.722 - 0.417i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.38 - 4.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.54 - 2.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.35 - 1.35i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.70 + 1.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.71iT - 71T^{2} \)
73 \( 1 + (-2.77 - 4.80i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.82 - 4.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.08 - 4.09i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.45 + 4.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.17258609372607498574640942859, −10.19538630500343956909169791404, −9.427012937732687601939550905064, −7.82859490229948925316218870416, −7.26113832672098255511675086249, −6.35772598977504819725254753395, −5.90526604242394001435035792538, −4.38614725881091652786373250714, −3.83816859603610912590880825124, −1.83844424034491858632480596595, 0.52184033722072633339037696164, 2.06416276444549104295069538284, 3.63391217077533522018641456295, 4.89748275005127703910243085086, 5.43981792739566726334314837847, 6.15047778404735721684040542045, 7.56029074988935003590720175678, 8.966035235733287827989506852545, 9.265812149382682611333048552174, 10.44692093565128369055110377961

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