Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.871 - 0.491i$
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 + (−0.995 − 1.41i)3-s + (0.499 + 0.866i)4-s − 5-s + (−0.153 − 1.72i)6-s + (−2.36 − 1.19i)7-s + 0.999i·8-s + (−1.01 + 2.82i)9-s + (−0.866 − 0.5i)10-s + 3.17i·11-s + (0.729 − 1.57i)12-s + (−1.82 − 1.05i)13-s + (−1.44 − 2.21i)14-s + (0.995 + 1.41i)15-s + (−0.5 + 0.866i)16-s + (−0.0900 + 0.155i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.574 − 0.818i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.0627 − 0.704i)6-s + (−0.892 − 0.451i)7-s + 0.353i·8-s + (−0.339 + 0.940i)9-s + (−0.273 − 0.158i)10-s + 0.957i·11-s + (0.210 − 0.453i)12-s + (−0.505 − 0.292i)13-s + (−0.386 − 0.592i)14-s + (0.257 + 0.365i)15-s + (−0.125 + 0.216i)16-s + (−0.0218 + 0.0378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.871 - 0.491i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ -0.871 - 0.491i)\)
\(L(1)\)  \(\approx\)  \(0.0865733 + 0.329866i\)
\(L(\frac12)\)  \(\approx\)  \(0.0865733 + 0.329866i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.995 + 1.41i)T \)
5 \( 1 + T \)
7 \( 1 + (2.36 + 1.19i)T \)
good11 \( 1 - 3.17iT - 11T^{2} \)
13 \( 1 + (1.82 + 1.05i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0900 - 0.155i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.17 - 2.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.789iT - 23T^{2} \)
29 \( 1 + (6.84 - 3.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.40 - 4.85i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.27 - 7.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.85 + 10.1i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.84 + 3.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.04 - 1.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.613 - 0.353i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.88 + 10.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.50 - 1.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.97 + 8.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (5.09 + 2.93i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.50 - 9.53i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.41 - 7.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.91 - 5.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.79 - 2.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.00527309232623611620363616488, −10.37733922477279785429600673826, −9.136709925901029822703103071086, −7.891306626378443883346329384104, −7.22603529473501737086816661249, −6.59666470887868777596812121333, −5.61711169095638953436008125563, −4.58153279708570722050397455031, −3.46738774747858567922758998432, −2.00792469822592582077986616024, 0.15160374395279728179089100916, 2.61217737992972012800478535011, 3.69938033725443493483271419735, 4.45522073447857044136695931123, 5.71030542538804687624495116621, 6.18660217178646871652809822014, 7.38844652298461647560523503845, 8.883431497623081397925226023827, 9.439511802746855844383469354953, 10.41014914064383288727016111000

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