Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.281 - 0.959i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + (−0.866 − 1.5i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−1.5 + 0.866i)6-s + (−2.5 − 0.866i)7-s + i·8-s + (−1.5 + 2.59i)9-s + (−0.866 + 0.5i)10-s + (1.09 + 0.633i)11-s + (0.866 + 1.5i)12-s + (−3 − 1.73i)13-s + (−0.866 + 2.5i)14-s + (−0.866 + 1.5i)15-s + 16-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.499 − 0.866i)3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.612 + 0.353i)6-s + (−0.944 − 0.327i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.273 + 0.158i)10-s + (0.331 + 0.191i)11-s + (0.249 + 0.433i)12-s + (−0.832 − 0.480i)13-s + (−0.231 + 0.668i)14-s + (−0.223 + 0.387i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.281 - 0.959i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (131, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.281 - 0.959i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + iT \)
3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good11 \( 1 + (-1.09 - 0.633i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.09 - 2.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.19 - 4.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.401 - 0.232i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.19iT - 31T^{2} \)
37 \( 1 + (-2.09 + 3.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.59 - 6.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + (9.29 - 5.36i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.19T + 59T^{2} \)
61 \( 1 - 0.928iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 1.26iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 16.5T + 79T^{2} \)
83 \( 1 + (0.401 + 0.696i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.19 - 14.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.3 + 7.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.924767300285042008729476236609, −9.316400136711598590675277256039, −7.936163045557615542462880159866, −7.41877468704941256557466854515, −6.18569606829290761853384528277, −5.36051672880127327013806407837, −4.09414946486629084257041971596, −2.90615923394850078885018616289, −1.48673456284610144314278423914, 0, 2.87972274484168293399399826024, 3.98810345368101453656271810666, 4.91080575512478723561876561556, 6.07754067367631429430677093575, 6.55035491725307089210655902052, 7.67797162887753511526762695191, 8.819501350374797024869512041280, 9.696976163937101526538702071040, 10.04459440084626268842410801047

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