Properties

Degree $2$
Conductor $630$
Sign $0.968 + 0.250i$
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 + 13-s + (0.499 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (1.5 + 2.59i)19-s + 0.999·20-s + 0.999·22-s + (3.5 + 6.06i)23-s + (−0.499 + 0.866i)25-s + (−0.5 − 0.866i)26-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 + 0.277·13-s + (0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.344 + 0.596i)19-s + 0.223·20-s + 0.213·22-s + (0.729 + 1.26i)23-s + (−0.0999 + 0.173i)25-s + (−0.0980 − 0.169i)26-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: $0.968 + 0.250i$
Motivic weight: \(1\)
Character: $\chi_{630} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26016 - 0.160587i\)
\(L(\frac12)\) \(\approx\) \(1.26016 - 0.160587i\)
\(L(\frac{3}{2})\) 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 \( 1 - T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.5 - 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.72413962993683057602993174288, −9.579244728233134039099603337953, −8.938693709671863553690117432568, −8.065587700764027843232734547504, −7.35744104645292134380895109596, −5.84806422441936973010158090255, −4.94290701192483883303318433442, −3.87201001363945203039859530711, −2.54682111093608289868636870610, −1.26631601636406220354746746157, 0.988148358679682787051604993347, 2.82452359752624132613041189539, 4.29676874573128115283301916888, 5.10031771781455566574995644038, 6.41230963520796542499914952544, 7.06721290904608414378550394850, 8.057337288023372939254638159756, 8.613731162274783637990180096929, 9.762326456317124792704289626831, 10.70283156795102607203305589936

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