Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.994 + 0.108i$
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 + (2.09 − 1.62i)7-s − 0.999·8-s + (−2.59 − 1.5i)11-s − 2.44·13-s + (2.44 + 0.999i)14-s + (−0.5 − 0.866i)16-s + (−0.878 − 0.507i)17-s + (0.878 − 0.507i)19-s − 3i·22-s + (2.12 + 3.67i)23-s + (−1.22 − 2.12i)26-s + (0.358 + 2.62i)28-s + 1.24i·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.790 − 0.612i)7-s − 0.353·8-s + (−0.783 − 0.452i)11-s − 0.679·13-s + (0.654 + 0.267i)14-s + (−0.125 − 0.216i)16-s + (−0.213 − 0.123i)17-s + (0.201 − 0.116i)19-s − 0.639i·22-s + (0.442 + 0.766i)23-s + (−0.240 − 0.416i)26-s + (0.0677 + 0.495i)28-s + 0.230i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.994 + 0.108i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.994 + 0.108i)$
$L(1)$  $\approx$  $2.021456402$
$L(\frac12)$  $\approx$  $2.021456402$
$L(\frac{3}{2})$   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 \( 1 + (-2.09 + 1.62i)T \)
good11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + (0.878 + 0.507i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.878 + 0.507i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.12 - 3.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.24iT - 29T^{2} \)
31 \( 1 + (-4.86 - 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.13 + 4.12i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 + 8.24iT - 43T^{2} \)
47 \( 1 + (-0.878 + 0.507i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.621 + 1.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.76 + 9.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.12 + 2.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 - 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-4.18 + 7.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.62 + 9.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.16iT - 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.76T + 97T^{2} \)
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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

−8.441939236013342183232629020959, −7.77754495055403280369455936370, −7.27592677301553890381451125440, −6.48044358924271194192160944862, −5.40806200411369790685704353832, −5.00924228799953951710097394554, −4.13302382709088214320641945190, −3.19233590744202983588730701914, −2.14345395145789943621590475316, −0.63684614533676459896866628468, 1.07181392900975254838991873451, 2.40728125170721828775730007720, 2.69066527785227089840014339626, 4.16977149879214274631260438910, 4.75661668803651793189988258412, 5.43648378769008941265248057609, 6.26310410642324739631359283171, 7.27107083840131855312058733498, 8.081143975961691164076867659338, 8.636287439451338958018552153677

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