Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.616 + 0.787i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (1.72 + 0.115i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−1.72 − 0.115i)6-s + (1.23 − 2.33i)7-s − 8-s + (2.97 + 0.400i)9-s + (0.5 + 0.866i)10-s + (1.85 − 3.21i)11-s + (1.72 + 0.115i)12-s + (−1.26 + 2.19i)13-s + (−1.23 + 2.33i)14-s + (−0.763 − 1.55i)15-s + 16-s + (−1.48 − 2.56i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.997 + 0.0669i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.705 − 0.0473i)6-s + (0.468 − 0.883i)7-s − 0.353·8-s + (0.991 + 0.133i)9-s + (0.158 + 0.273i)10-s + (0.559 − 0.969i)11-s + (0.498 + 0.0334i)12-s + (−0.351 + 0.609i)13-s + (−0.331 + 0.624i)14-s + (−0.197 − 0.401i)15-s + 0.250·16-s + (−0.359 − 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.616 + 0.787i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (121, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 630,\ (\ :1/2),\ 0.616 + 0.787i)$
$L(1)$  $\approx$  $1.37775 - 0.671326i$
$L(\frac12)$  $\approx$  $1.37775 - 0.671326i$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 + (-1.72 - 0.115i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.23 + 2.33i)T \)
good11 \( 1 + (-1.85 + 3.21i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.26 - 2.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.48 + 2.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.26 - 2.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.21 + 2.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.04 + 8.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.87T + 31T^{2} \)
37 \( 1 + (-1.36 + 2.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.50 - 9.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.65 - 2.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.695T + 47T^{2} \)
53 \( 1 + (-4.25 - 7.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 2.43T + 71T^{2} \)
73 \( 1 + (-0.729 - 1.26i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 2.27T + 79T^{2} \)
83 \( 1 + (5.39 + 9.34i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.50 - 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.0494 - 0.0856i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.15873907014077142026975223008, −9.576667381982433317111651261252, −8.578746602768917489279994557326, −8.095272149313059941250851168663, −7.23010102292413917914135299022, −6.27171831306316953364707690763, −4.58240243746109611402564578949, −3.81105826471799135147906750983, −2.41367702432752893146714146140, −1.02566870571423921519659894992, 1.75074932614824421364297080782, 2.65802543077249008264480254044, 3.90809011931793861684018202564, 5.21363800798102389169561134418, 6.67198591147576454197958399140, 7.32057537822063420140557243228, 8.332019744585667318476820031561, 8.810661982012815438844999107036, 9.767556862267773803399412230589, 10.42948095717764522513083791080

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