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Properties

Label 2-630-63.38-c1-0-21
Degree $2$
Conductor $630$
Sign $0.879 - 0.476i$
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  + i·2-s + (1.51 − 0.835i)3-s − 4-s + (−0.5 + 0.866i)5-s + (0.835 + 1.51i)6-s + (2.64 − 0.162i)7-s − i·8-s + (1.60 − 2.53i)9-s + (−0.866 − 0.5i)10-s + (−0.650 + 0.375i)11-s + (−1.51 + 0.835i)12-s + (2.72 − 1.57i)13-s + (0.162 + 2.64i)14-s + (−0.0350 + 1.73i)15-s + 16-s + (−0.433 + 0.750i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.875 − 0.482i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (0.341 + 0.619i)6-s + (0.998 − 0.0612i)7-s − 0.353i·8-s + (0.534 − 0.845i)9-s + (−0.273 − 0.158i)10-s + (−0.196 + 0.113i)11-s + (−0.437 + 0.241i)12-s + (0.754 − 0.435i)13-s + (0.0433 + 0.705i)14-s + (−0.00906 + 0.447i)15-s + 0.250·16-s + (−0.105 + 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.879 - 0.476i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.879 - 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00404 + 0.508094i\)
\(L(\frac12)\) \(\approx\) \(2.00404 + 0.508094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.51 + 0.835i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.64 + 0.162i)T \)
good11 \( 1 + (0.650 - 0.375i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.72 + 1.57i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.433 - 0.750i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.23 + 0.713i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.38 - 2.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.23 + 3.02i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.21iT - 31T^{2} \)
37 \( 1 + (-3.60 - 6.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.08 - 3.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.69 + 6.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.45T + 47T^{2} \)
53 \( 1 + (-0.790 - 0.456i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.43T + 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 - 5.43iT - 71T^{2} \)
73 \( 1 + (-0.539 - 0.311i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.00585T + 79T^{2} \)
83 \( 1 + (1.01 - 1.76i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.22 - 2.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.23 + 5.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.64392294132513511871773215366, −9.503733617378186934632276903496, −8.645017299354882896703099349350, −7.933407054171278801636446545779, −7.34275930191820369834771796709, −6.39468324513770242643961168541, −5.23363005137765733925478776092, −4.06496259987245837946743216121, −2.97160993129776856699970216028, −1.39972455419111218896668610647, 1.45532418289624581661529200543, 2.66674522722353249703267442113, 3.91737200283510735521110161904, 4.60959816154095756875958127586, 5.64360906152233060668068619515, 7.39361783613270393695010212910, 8.133867940527420781583047679256, 8.972981911075646381681482976288, 9.450729719135608113754965019418, 10.79566227824426069867444340306

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