Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
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

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.968 - 0.250i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (541, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.968 - 0.250i)\)
\(L(1)\)  \(\approx\)  \(1.26016 + 0.160587i\)
\(L(\frac12)\)  \(\approx\)  \(1.26016 + 0.160587i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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