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Properties

Label 2-630-105.89-c1-0-2
Degree $2$
Conductor $630$
Sign $0.204 - 0.978i$
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 + (−0.499 − 0.866i)4-s + (0.104 + 2.23i)5-s + (−1.39 + 2.24i)7-s − 0.999·8-s + (1.98 + 1.02i)10-s + (−1.37 + 0.796i)11-s − 0.925·13-s + (1.24 + 2.33i)14-s + (−0.5 + 0.866i)16-s + (−3.42 + 1.97i)17-s + (0.541 + 0.312i)19-s + (1.88 − 1.20i)20-s + 1.59i·22-s + (−3.89 + 6.74i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0467 + 0.998i)5-s + (−0.528 + 0.848i)7-s − 0.353·8-s + (0.628 + 0.324i)10-s + (−0.415 + 0.240i)11-s − 0.256·13-s + (0.332 + 0.623i)14-s + (−0.125 + 0.216i)16-s + (−0.831 + 0.479i)17-s + (0.124 + 0.0717i)19-s + (0.420 − 0.269i)20-s + 0.339i·22-s + (−0.811 + 1.40i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.204 - 0.978i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881750 + 0.716920i\)
\(L(\frac12)\) \(\approx\) \(0.881750 + 0.716920i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.104 - 2.23i)T \)
7 \( 1 + (1.39 - 2.24i)T \)
good11 \( 1 + (1.37 - 0.796i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.925T + 13T^{2} \)
17 \( 1 + (3.42 - 1.97i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.541 - 0.312i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.89 - 6.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.34iT - 29T^{2} \)
31 \( 1 + (-8.94 + 5.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.369 - 0.213i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.35T + 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 + (2.40 + 1.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.67 - 2.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.10 + 5.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.52 - 5.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.308 + 0.178i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.07iT - 71T^{2} \)
73 \( 1 + (-3.41 - 5.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.52 - 7.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.809iT - 83T^{2} \)
89 \( 1 + (2.00 - 3.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.87T + 97T^{2} \)
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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

−10.83568508968723935626977784846, −9.999121063538378027113591901627, −9.363693214090708507462525952295, −8.229064938569575298409308592155, −7.09098354341946073861577449369, −6.16780478519359044350955054717, −5.35489626909504430867666616139, −3.99253037771462267217204164924, −2.94590402180687852083730840689, −2.07982881987002552131539045925, 0.53036340838371623308546975650, 2.64941858666161446478451001158, 4.19905597377033991877013721282, 4.68976066862255821232744434909, 5.95262204329433404989781290699, 6.71366422143641325394967878700, 7.83383932301243355256185619737, 8.446496054084459241517153701806, 9.512454466489752305575865545967, 10.20587487756383953957899751379

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