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Properties

Label 2-630-7.2-c1-0-2
Degree $2$
Conductor $630$
Sign $0.968 - 0.250i$
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 + (−0.5 + 0.866i)5-s + (−2 + 1.73i)7-s − 0.999·8-s + (0.499 + 0.866i)10-s + (1.5 + 2.59i)11-s + 5·13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (0.5 − 0.866i)19-s + 0.999·20-s + 3·22-s + (1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (0.158 + 0.273i)10-s + (0.452 + 0.783i)11-s + 1.38·13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (0.114 − 0.198i)19-s + 0.223·20-s + 0.639·22-s + (0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56214 + 0.199069i\)
\(L(\frac12)\) \(\approx\) \(1.56214 + 0.199069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.44818051785564739078228122799, −10.12612932223358285627844473903, −8.905848776088895795899535002048, −8.279904560490615960648860263796, −6.70694526413765678264311412875, −6.23591505780769908319746527635, −4.99887129157353311866874898179, −3.76805112712822206061347838175, −3.02350979959323521801885611695, −1.53824127778636840542222600082, 0.871158682691370579625836974050, 3.24024604742150131386801335067, 3.87479515698245061653802027953, 5.14995645061735800448202820056, 6.10542688645873288368260961912, 6.89761810652307725636640280718, 7.85243620877010769439574829020, 8.744832402103173525786839833692, 9.502503365720733073728977407072, 10.56419179604024261429274951854

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