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Properties

Label 2-630-63.25-c1-0-7
Degree $2$
Conductor $630$
Sign $0.839 - 0.543i$
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  − 2-s + (−0.879 + 1.49i)3-s + 4-s + (0.5 − 0.866i)5-s + (0.879 − 1.49i)6-s + (2.58 + 0.571i)7-s − 8-s + (−1.45 − 2.62i)9-s + (−0.5 + 0.866i)10-s + (−0.0955 − 0.165i)11-s + (−0.879 + 1.49i)12-s + (2.58 + 4.48i)13-s + (−2.58 − 0.571i)14-s + (0.852 + 1.50i)15-s + 16-s + (2.72 − 4.71i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.507 + 0.861i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.358 − 0.609i)6-s + (0.976 + 0.215i)7-s − 0.353·8-s + (−0.484 − 0.874i)9-s + (−0.158 + 0.273i)10-s + (−0.0288 − 0.0499i)11-s + (−0.253 + 0.430i)12-s + (0.718 + 1.24i)13-s + (−0.690 − 0.152i)14-s + (0.220 + 0.389i)15-s + 0.250·16-s + (0.660 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02846 + 0.303574i\)
\(L(\frac12)\) \(\approx\) \(1.02846 + 0.303574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.879 - 1.49i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.58 - 0.571i)T \)
good11 \( 1 + (0.0955 + 0.165i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.58 - 4.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.72 + 4.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.06 + 3.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.25 + 7.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.67 - 6.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.19T + 31T^{2} \)
37 \( 1 + (-4.44 - 7.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.65 - 6.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.180 - 0.312i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 + (3.12 - 5.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.01T + 59T^{2} \)
61 \( 1 - 2.86T + 61T^{2} \)
67 \( 1 - 5.95T + 67T^{2} \)
71 \( 1 + 9.49T + 71T^{2} \)
73 \( 1 + (2.71 - 4.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.95T + 79T^{2} \)
83 \( 1 + (-1.58 + 2.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.43 + 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.26 + 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.74007122387693482580969142629, −9.708096684886395140727346661867, −8.942889451604628798350500800269, −8.483889364523876731866023012777, −7.10423663629973366014235420284, −6.19428831061171921462552579604, −5.03814442909899807418031543289, −4.40101969530208219067520010871, −2.76623168459772470294465969679, −1.11237675984058315613027233949, 1.08445874562826869742992800865, 2.15244198884246532912923334627, 3.71710358557458617529038449292, 5.54609966683476541845990852736, 5.92177025514585630740946101276, 7.22946005936920000090850059305, 7.888865064085092300699281035362, 8.423547900357202520312672503320, 9.797169822246587587385291841296, 10.79855493696796678247266399921

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