Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.623 + 0.781i$
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 + (−1.51 − 0.832i)3-s + (0.499 − 0.866i)4-s − 5-s + (−1.73 + 0.0386i)6-s + (2.13 − 1.55i)7-s − 0.999i·8-s + (1.61 + 2.52i)9-s + (−0.866 + 0.5i)10-s + 0.450i·11-s + (−1.48 + 0.899i)12-s + (4.26 − 2.46i)13-s + (1.07 − 2.41i)14-s + (1.51 + 0.832i)15-s + (−0.5 − 0.866i)16-s + (−3.93 − 6.81i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.876 − 0.480i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.706 + 0.0157i)6-s + (0.807 − 0.589i)7-s − 0.353i·8-s + (0.538 + 0.842i)9-s + (−0.273 + 0.158i)10-s + 0.135i·11-s + (−0.427 + 0.259i)12-s + (1.18 − 0.682i)13-s + (0.286 − 0.646i)14-s + (0.392 + 0.214i)15-s + (−0.125 − 0.216i)16-s + (−0.953 − 1.65i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.623 + 0.781i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (551, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ -0.623 + 0.781i)\)
\(L(1)\)  \(\approx\)  \(0.607555 - 1.26205i\)
\(L(\frac12)\)  \(\approx\)  \(0.607555 - 1.26205i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.51 + 0.832i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.13 + 1.55i)T \)
good11 \( 1 - 0.450iT - 11T^{2} \)
13 \( 1 + (-4.26 + 2.46i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.93 + 6.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.75 + 2.74i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.86iT - 23T^{2} \)
29 \( 1 + (8.05 + 4.65i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.497 - 0.287i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.721 - 1.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.956 - 1.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.459 - 0.795i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.71 - 6.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.30 + 4.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.43 + 7.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.54 + 4.93i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.32 + 4.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.88iT - 71T^{2} \)
73 \( 1 + (-4.91 + 2.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.76 - 8.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.98 - 3.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.69 + 5.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.88124546204190720786173676396, −9.653825293233536471622610654666, −8.342655434213806663355976228719, −7.40964565235914257129361322838, −6.67572850425944067420481333303, −5.55516820977786580455550996329, −4.73494701927000330121221789011, −3.83404032822704120468827568085, −2.17287556725930561222134434591, −0.71846857834881256163283289561, 1.87395014966513309745371255815, 3.91676130035902997563121790055, 4.25558961958222668838317349814, 5.55499490119568772901604393406, 6.16670974006524833896191066138, 7.07183732303164270897198052877, 8.561627348188623079118254795191, 8.727604183440050957509723117493, 10.50758240249912062957117282896, 10.97509457346352408952652667445

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