Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.537 + 0.843i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (1.35 − 1.08i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−1.35 + 1.08i)6-s + (2.32 + 1.26i)7-s − 8-s + (0.653 − 2.92i)9-s + (0.5 + 0.866i)10-s + (−0.975 + 1.68i)11-s + (1.35 − 1.08i)12-s + (2.18 − 3.77i)13-s + (−2.32 − 1.26i)14-s + (−1.61 − 0.629i)15-s + 16-s + (2.22 + 3.85i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.780 − 0.625i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.551 + 0.442i)6-s + (0.878 + 0.478i)7-s − 0.353·8-s + (0.217 − 0.975i)9-s + (0.158 + 0.273i)10-s + (−0.294 + 0.509i)11-s + (0.390 − 0.312i)12-s + (0.605 − 1.04i)13-s + (−0.621 − 0.338i)14-s + (−0.416 − 0.162i)15-s + 0.250·16-s + (0.539 + 0.934i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.537 + 0.843i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (121, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 630,\ (\ :1/2),\ 0.537 + 0.843i)$
$L(1)$  $\approx$  $1.31724 - 0.722948i$
$L(\frac12)$  $\approx$  $1.31724 - 0.722948i$
$L(\frac{3}{2})$   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 \( 1 + (-1.35 + 1.08i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.32 - 1.26i)T \)
good11 \( 1 + (0.975 - 1.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.18 + 3.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.22 - 3.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.18 + 3.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.957 + 1.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.725 - 1.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.78T + 31T^{2} \)
37 \( 1 + (1.44 - 2.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.07 + 7.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.39 + 2.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.79T + 47T^{2} \)
53 \( 1 + (3.07 + 5.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.10T + 59T^{2} \)
61 \( 1 - 0.523T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + (-7.69 - 13.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.55T + 79T^{2} \)
83 \( 1 + (-1.48 - 2.58i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.62 + 8.01i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.19 + 10.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.34520962433208209868902261642, −9.372998917200258911859462379741, −8.433046772779006982255959324400, −8.114823553583234689798136388628, −7.28229580192691099071954782632, −6.11762027960766285675072163798, −4.99060973071115166459906690874, −3.49818007869526484475510014641, −2.28239644659945936398164086706, −1.10280953856654043222127451424, 1.55854703639707183198003228307, 2.96714645006397097404559514240, 3.98580701003944981584742847505, 5.10280951345298611155099028782, 6.46310174191804212110827317160, 7.79590820884555246943163064974, 7.904771313927550612447669050311, 9.081336250234452506562289260524, 9.753425576088030572012755866705, 10.67593615410207951073046366657

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