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Properties

Label 2-630-7.4-c1-0-3
Degree $2$
Conductor $630$
Sign $-0.605 - 0.795i$
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 + (0.5 + 2.59i)7-s − 0.999·8-s + (0.499 − 0.866i)10-s + (−1 + 1.73i)11-s + (−2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (3 + 5.19i)19-s + 0.999·20-s − 1.99·22-s + (1.5 + 2.59i)23-s + (−0.499 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.188 + 0.981i)7-s − 0.353·8-s + (0.158 − 0.273i)10-s + (−0.301 + 0.522i)11-s + (−0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (0.688 + 1.19i)19-s + 0.223·20-s − 0.426·22-s + (0.312 + 0.541i)23-s + (−0.0999 + 0.173i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625952 + 1.26269i\)
\(L(\frac12)\) \(\approx\) \(0.625952 + 1.26269i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
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 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
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

−11.08369563708178282558111522483, −9.774994158027101588594055804488, −9.082381216445058600516055857688, −8.086609108709684876163148827427, −7.54365967889550346917120599562, −6.20016047228734992527637508116, −5.52829800016648566272453472331, −4.57043987330343540565749871491, −3.45034626536074867160129536224, −1.93748209109471200044832615615, 0.69351362280457246969452760796, 2.49110987359881786934388463418, 3.54244130032619760571013131888, 4.54816319253224974207564139304, 5.50623407330431181851177413022, 6.84842748572491860166344025626, 7.45965657095936610689018803106, 8.692042187570068158538691971701, 9.591262146566336829551694601789, 10.54411253733305747923782823442

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