Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.398 + 0.917i$
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 + (−1.28 + 1.16i)3-s + (0.499 + 0.866i)4-s − 5-s + (−1.69 + 0.361i)6-s + (−0.555 − 2.58i)7-s + 0.999i·8-s + (0.307 − 2.98i)9-s + (−0.866 − 0.5i)10-s − 4.07i·11-s + (−1.64 − 0.533i)12-s + (−5.20 − 3.00i)13-s + (0.812 − 2.51i)14-s + (1.28 − 1.16i)15-s + (−0.5 + 0.866i)16-s + (−0.641 + 1.11i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.742 + 0.669i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.691 + 0.147i)6-s + (−0.209 − 0.977i)7-s + 0.353i·8-s + (0.102 − 0.994i)9-s + (−0.273 − 0.158i)10-s − 1.22i·11-s + (−0.475 − 0.154i)12-s + (−1.44 − 0.832i)13-s + (0.217 − 0.672i)14-s + (0.332 − 0.299i)15-s + (−0.125 + 0.216i)16-s + (−0.155 + 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.398 + 0.917i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.398 + 0.917i)\)
\(L(1)\)  \(\approx\)  \(0.749407 - 0.491330i\)
\(L(\frac12)\)  \(\approx\)  \(0.749407 - 0.491330i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.28 - 1.16i)T \)
5 \( 1 + T \)
7 \( 1 + (0.555 + 2.58i)T \)
good11 \( 1 + 4.07iT - 11T^{2} \)
13 \( 1 + (5.20 + 3.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.641 - 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.90 + 1.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.18iT - 23T^{2} \)
29 \( 1 + (-6.21 + 3.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.79 - 3.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.90 + 5.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.36 + 2.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.43 + 4.22i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.13 + 7.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.48 + 4.32i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.630 + 1.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.56 + 1.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.31 - 7.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + (6.02 + 3.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.09 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.87 - 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.29 + 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.8 - 9.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.58457137180192881173801034711, −9.801755968662116440963268495521, −8.624472350398983300873177493194, −7.50377005714802818214323346587, −6.82977208650504301494512955015, −5.66273527805387917472218990619, −4.97696281364427852693587530036, −3.89965331093658811032033543278, −3.12950281233084874325303038163, −0.42596909628755233020467051758, 1.80316904556372218440248277197, 2.79921769398120015236735029410, 4.57768519287402911526685693228, 5.05540042410932620847340304550, 6.25275983427477600878824767112, 7.05395408076513446067247247291, 7.81088729019463929887159141341, 9.249853622161305973674672522448, 9.980316472792767623587930171219, 11.07559420243131439794482190874

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