Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.776 - 0.629i$
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 + (1.64 − 0.542i)3-s + (0.499 + 0.866i)4-s − 5-s + (1.69 + 0.352i)6-s + (−0.673 + 2.55i)7-s + 0.999i·8-s + (2.41 − 1.78i)9-s + (−0.866 − 0.5i)10-s − 1.46i·11-s + (1.29 + 1.15i)12-s + (6.03 + 3.48i)13-s + (−1.86 + 1.87i)14-s + (−1.64 + 0.542i)15-s + (−0.5 + 0.866i)16-s + (3.30 − 5.72i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.949 − 0.313i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.692 + 0.143i)6-s + (−0.254 + 0.967i)7-s + 0.353i·8-s + (0.803 − 0.594i)9-s + (−0.273 − 0.158i)10-s − 0.441i·11-s + (0.373 + 0.332i)12-s + (1.67 + 0.966i)13-s + (−0.497 + 0.502i)14-s + (−0.424 + 0.140i)15-s + (−0.125 + 0.216i)16-s + (0.801 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.776 - 0.629i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.776 - 0.629i)\)
\(L(1)\)  \(\approx\)  \(2.58524 + 0.916127i\)
\(L(\frac12)\)  \(\approx\)  \(2.58524 + 0.916127i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.64 + 0.542i)T \)
5 \( 1 + T \)
7 \( 1 + (0.673 - 2.55i)T \)
good11 \( 1 + 1.46iT - 11T^{2} \)
13 \( 1 + (-6.03 - 3.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.30 + 5.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.08 - 1.78i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.36iT - 23T^{2} \)
29 \( 1 + (-1.49 + 0.862i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.79 - 3.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.75 + 4.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.632 - 1.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.24 + 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.69 + 4.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.60 + 3.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.746 + 1.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.23 - 1.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.27 + 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 + (1.11 + 0.640i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.497 - 0.860i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.93 - 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.18 + 3.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.04 - 5.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.89291306048674395343049997952, −9.405323550665571544944706368124, −8.850928799418149127185744720949, −8.078325142055364604890892230353, −7.11904411781177625666640726103, −6.26017953192255815136541793329, −5.25141132948989630010552387418, −3.78288867888653693366980170773, −3.25412866591335464441664142080, −1.81243144575432624489488522833, 1.38958634286749851105565489562, 3.02173729036495188030799979396, 3.83850735219935869419544747394, 4.45172806593322539002944641887, 5.93333644467038054273575447883, 6.95958450295455120234025257639, 8.038744210946046208905067676110, 8.597885051560082232286449129046, 9.900336487514425785394423922400, 10.60451978284951982156749581650

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