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  :  $\chi_{630} (551, \cdot )$
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

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

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