Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.558 + 0.829i$
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 + (0.499 − 0.866i)4-s + (−2.24 + 1.39i)7-s + 0.999i·8-s + (1.37 + 0.796i)11-s − 0.925i·13-s + (1.24 − 2.33i)14-s + (−0.5 − 0.866i)16-s + (1.97 − 3.42i)17-s + (−0.541 + 0.312i)19-s − 1.59·22-s + (−6.74 + 3.89i)23-s + (0.462 + 0.801i)26-s + (0.0890 + 2.64i)28-s − 9.34i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.848 + 0.528i)7-s + 0.353i·8-s + (0.415 + 0.240i)11-s − 0.256i·13-s + (0.332 − 0.623i)14-s + (−0.125 − 0.216i)16-s + (0.479 − 0.831i)17-s + (−0.124 + 0.0717i)19-s − 0.339·22-s + (−1.40 + 0.811i)23-s + (0.0907 + 0.157i)26-s + (0.0168 + 0.499i)28-s − 1.73i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.558 + 0.829i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.558 + 0.829i)$
$L(1)$  $\approx$  $0.7869755927$
$L(\frac12)$  $\approx$  $0.7869755927$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.24 - 1.39i)T \)
good11 \( 1 + (-1.37 - 0.796i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.925iT - 13T^{2} \)
17 \( 1 + (-1.97 + 3.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.541 - 0.312i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.74 - 3.89i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.34iT - 29T^{2} \)
31 \( 1 + (-8.94 - 5.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.213 + 0.369i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.35T + 41T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + (1.39 + 2.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.90 + 1.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.10 - 5.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.52 + 5.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.178 + 0.308i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.07iT - 71T^{2} \)
73 \( 1 + (-5.91 - 3.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.52 - 7.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.809T + 83T^{2} \)
89 \( 1 + (2.00 + 3.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.87iT - 97T^{2} \)
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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

−8.421037360299255376474633865984, −8.009111539579399325583136295526, −7.00046420994229076714313948392, −6.41515895432408134543492577626, −5.72802167430087252222100277221, −4.88066502129699075544300699441, −3.73604385739763744654839646187, −2.82512595225820125023508987380, −1.78697066530437441568778080714, −0.35739825557514583431877341096, 0.971630172883152419548890067614, 2.09966350027762316649076791840, 3.25453379199927176337296928766, 3.84317013103081627319001619529, 4.80474594541315235136142232940, 6.14748322090167156194785859207, 6.50110302121535567941376699095, 7.36636992716393573730113771008, 8.267255191855166654062890874815, 8.707646229340284736895282445318

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