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  :  $\chi_{630} (311, \cdot )$
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

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

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