Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.926 - 0.376i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.62 − 0.295i)7-s − 0.999·8-s + (−0.570 − 0.329i)11-s − 6.13·13-s + (−1.05 − 2.42i)14-s + (−0.5 − 0.866i)16-s + (−4.22 − 2.43i)17-s + (6.30 − 3.63i)19-s − 0.659i·22-s + (2.29 + 3.98i)23-s + (−3.06 − 5.31i)26-s + (1.57 − 2.12i)28-s + 8.09i·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.993 − 0.111i)7-s − 0.353·8-s + (−0.172 − 0.0993i)11-s − 1.70·13-s + (−0.282 − 0.648i)14-s + (−0.125 − 0.216i)16-s + (−1.02 − 0.591i)17-s + (1.44 − 0.834i)19-s − 0.140i·22-s + (0.479 + 0.830i)23-s + (−0.601 − 1.04i)26-s + (0.296 − 0.402i)28-s + 1.50i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.926 - 0.376i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.926 - 0.376i)$
$L(1)$  $\approx$  $1.442330855$
$L(\frac12)$  $\approx$  $1.442330855$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.62 + 0.295i)T \)
good11 \( 1 + (0.570 + 0.329i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.13T + 13T^{2} \)
17 \( 1 + (4.22 + 2.43i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.30 + 3.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.29 - 3.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.09iT - 29T^{2} \)
31 \( 1 + (-0.759 - 0.438i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.75 + 5.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.25T + 41T^{2} \)
43 \( 1 - 9.03iT - 43T^{2} \)
47 \( 1 + (-10.3 + 6.00i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.10 + 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.06 + 7.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0618 - 0.0357i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.15 + 0.666i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.60iT - 71T^{2} \)
73 \( 1 + (1.41 - 2.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.88 - 5.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.44iT - 83T^{2} \)
89 \( 1 + (2.66 + 4.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−8.820172194833088083535530223822, −7.61587143993724180776914239368, −7.16753829255247471643621689618, −6.69092471209560247753820102176, −5.53597535347886156976065895685, −5.06811252663576507376740970921, −4.16398303820814398149408788652, −3.07518946488783913353619711615, −2.51357063567007004537444307144, −0.57258606620282186612627081130, 0.76232980936981706604574400503, 2.41075560443149823939953522538, 2.73379816948140992270377449877, 3.97863493283235588732720654084, 4.56730454698310873632846253298, 5.61034329150078486384611139452, 6.18680711685518099193761811450, 7.17331774376656340304828178644, 7.76812256452814450207970314900, 8.921204820129875443462150974457

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