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Properties

Label 2-630-63.58-c1-0-16
Degree $2$
Conductor $630$
Sign $0.994 - 0.101i$
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 + (−1.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (−1.5 + 0.866i)6-s + (1.62 − 2.09i)7-s + 8-s + (1.5 − 2.59i)9-s + (0.5 + 0.866i)10-s + (2.12 − 3.67i)11-s + (−1.5 + 0.866i)12-s + (−1 + 1.73i)13-s + (1.62 − 2.09i)14-s + (−1.5 − 0.866i)15-s + 16-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 + (0.612 − 0.790i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.158 + 0.273i)10-s + (0.639 − 1.10i)11-s + (−0.433 + 0.249i)12-s + (−0.277 + 0.480i)13-s + (0.433 − 0.558i)14-s + (−0.387 − 0.223i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.994 - 0.101i$
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: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.994 - 0.101i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00100 + 0.102045i\)
\(L(\frac12)\) \(\approx\) \(2.00100 + 0.102045i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 2.09i)T \)
good11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 + 1.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.24 - 7.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.621 + 1.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + (-4.12 + 7.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.24T + 47T^{2} \)
53 \( 1 + (0.878 + 1.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.75T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 6.48T + 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + (2.24 + 3.88i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.24 - 5.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.98761912161346295223022055487, −9.960842970289718686105183226863, −9.092249026951767363239033452112, −7.66524654443981009408579037433, −6.78837219976460915282642804045, −5.99491434951561075022110030736, −5.04683163232396621109634286138, −4.15859256428984410742948354755, −3.19472072902014304202356822692, −1.23657382790840338906751105788, 1.42410851180371216897370782545, 2.57737460756336344666404610470, 4.48972574018327275532394879511, 4.99944674627010406895746814650, 5.96824082078872918442288110566, 6.77999265447686058270460046657, 7.73956872900786788748859661733, 8.739805343827009627407135644200, 9.955329984378319973722336407024, 10.75105786525438043020986421527

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