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Properties

Label 2-630-63.58-c1-0-17
Degree $2$
Conductor $630$
Sign $-0.841 + 0.540i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (1.5 − 0.866i)6-s + (−0.5 + 2.59i)7-s − 8-s + (1.5 − 2.59i)9-s + (0.5 + 0.866i)10-s + (−1.5 + 0.866i)12-s + (−1 + 1.73i)13-s + (0.5 − 2.59i)14-s + (1.5 + 0.866i)15-s + 16-s + (−1.5 + 2.59i)18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (0.612 − 0.353i)6-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (0.5 − 0.866i)9-s + (0.158 + 0.273i)10-s + (−0.433 + 0.249i)12-s + (−0.277 + 0.480i)13-s + (0.133 − 0.694i)14-s + (0.387 + 0.223i)15-s + 0.250·16-s + (−0.353 + 0.612i)18-s + (−0.229 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.841 + 0.540i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
good11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (4 + 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08965789392874766315017603931, −9.441059882757633026448659941786, −8.729650244708553949583419633140, −7.68246697990669215026946088794, −6.54145691846282398273810138875, −5.77085980274682732295768288850, −4.81762924018274443763444759378, −3.57012788919793055736804976877, −1.91664370796084748685425521482, 0, 1.49592432160490729146114110591, 3.13670966998521605001242466226, 4.55081947928213342217778257302, 5.74803525457187469512994772055, 6.78787901492335549861798827823, 7.31789886586117032859422630741, 8.079672459695154435841726705262, 9.361469303294492760968238914149, 10.31588875935745795208413781780

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