Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.969 + 0.244i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.404 + 1.68i)3-s + (0.499 − 0.866i)4-s − 5-s + (−1.19 − 1.25i)6-s + (1.07 − 2.41i)7-s + 0.999i·8-s + (−2.67 + 1.36i)9-s + (0.866 − 0.5i)10-s − 4.51i·11-s + (1.66 + 0.491i)12-s + (1.92 − 1.11i)13-s + (0.279 + 2.63i)14-s + (−0.404 − 1.68i)15-s + (−0.5 − 0.866i)16-s + (−3.31 − 5.74i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.233 + 0.972i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.486 − 0.512i)6-s + (0.405 − 0.914i)7-s + 0.353i·8-s + (−0.890 + 0.454i)9-s + (0.273 − 0.158i)10-s − 1.36i·11-s + (0.479 + 0.141i)12-s + (0.534 − 0.308i)13-s + (0.0747 + 0.703i)14-s + (−0.104 − 0.434i)15-s + (−0.125 − 0.216i)16-s + (−0.803 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.969 + 0.244i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (551, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.969 + 0.244i)\)
\(L(1)\)  \(\approx\)  \(1.00920 - 0.125534i\)
\(L(\frac12)\)  \(\approx\)  \(1.00920 - 0.125534i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.404 - 1.68i)T \)
5 \( 1 + T \)
7 \( 1 + (-1.07 + 2.41i)T \)
good11 \( 1 + 4.51iT - 11T^{2} \)
13 \( 1 + (-1.92 + 1.11i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.31 + 5.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.71 - 3.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.02iT - 23T^{2} \)
29 \( 1 + (-1.08 - 0.623i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.32 + 2.49i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.475 - 0.822i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.31 - 7.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.87 + 8.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.53 - 4.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.68 - 2.70i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.63 - 2.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.88 + 5.70i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.76 - 4.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.57iT - 71T^{2} \)
73 \( 1 + (-1.11 + 0.642i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.98 - 3.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.71 + 9.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.89 + 15.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.24 + 4.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.58125815074774046921674582793, −9.621850235796343058037503192992, −8.803261006022252403250964599759, −8.060484247966980648957567809183, −7.30038685048582182753599716702, −6.00369612193734964296184061224, −5.03584171977958975755715702684, −3.94732847149666042163249947874, −2.93873023182403537914282895134, −0.71239264162939872530613925872, 1.48884495043448573995217388845, 2.41114888308787631302246261996, 3.74514513543747804932084830680, 5.20619237857371718603416607170, 6.42238663124447924874870822696, 7.34672457129428475852237017268, 7.967534651704417337248169691994, 8.959871366139901659320619391828, 9.411421249800275193280413546950, 10.85476320980654797749284963890

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