Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.857 + 0.514i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.857 + 0.514i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.857 + 0.514i)$
$L(1)$  $\approx$  $1.996527106$
$L(\frac12)$  $\approx$  $1.996527106$
$L(\frac{3}{2})$   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

−8.634149366557047152358417771090, −8.082686701502574520766898289560, −7.35911566843887051384716441012, −6.34791270036662440165793084782, −5.42837930242865657549590514567, −4.75183850597891000185660187399, −3.65240643684245874183247086261, −3.04915449528543827815258873717, −1.69772953364101584171485718885, −1.03442203938060976759014830724, 0.968290881919465620142290930233, 1.78194547685819135794397479003, 3.33157569326573480974808863424, 4.07040313458720723655921404859, 5.24051240851272780891176417807, 5.62879488163301911647397445365, 6.48600716315058795428665505794, 7.55102226754816735386828547206, 7.920727772307499854385796107855, 8.492324591539756026819258942508

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