Properties

Degree $2$
Conductor $630$
Sign $0.386 - 0.922i$
Motivic weight $1$
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 + (−0.5 + 0.866i)11-s + 7·13-s + (2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (−0.5 − 0.866i)19-s − 0.999·20-s − 0.999·22-s + (0.5 + 0.866i)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 + (−0.150 + 0.261i)11-s + 1.94·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (−0.114 − 0.198i)19-s − 0.223·20-s − 0.213·22-s + (0.104 + 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.386 - 0.922i$
Motivic weight: \(1\)
Character: $\chi_{630} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67027 + 1.11103i\)
\(L(\frac12)\) \(\approx\) \(1.67027 + 1.11103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 7T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−10.81969285282630764680299922057, −10.02248943322612552983349154037, −8.534316096540877480763042876204, −8.312231360735852369355269882641, −6.97359734140330362362674046285, −6.41903359710852254102492728028, −5.31175734659128630474313456349, −4.26974582223375506078480735217, −3.33724876461643906643390240718, −1.56088455202916376653480763358, 1.21355741267945689276250842244, 2.47701075094169862455421068176, 3.77626609043496592640883211206, 4.85809735754376959928951563893, 5.69958069259223045967488953392, 6.59300376078337202567710398881, 8.202568889945152015460403992989, 8.666585487367050046528870742845, 9.572632805190784027739284739037, 10.69801583489640963834149074405

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