Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.766 - 0.641i$
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 + (2.63 − 0.189i)7-s + 0.999·8-s + (−3.44 − 1.99i)11-s + 0.0681·13-s + (−1.48 − 2.19i)14-s + (−0.5 − 0.866i)16-s + (−6.34 − 3.66i)17-s + (1.76 − 1.01i)19-s + 3.98i·22-s + (1.86 + 3.23i)23-s + (−0.0340 − 0.0590i)26-s + (−1.15 + 2.38i)28-s + 0.898i·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.997 − 0.0716i)7-s + 0.353·8-s + (−1.03 − 0.600i)11-s + 0.0189·13-s + (−0.396 − 0.585i)14-s + (−0.125 − 0.216i)16-s + (−1.53 − 0.888i)17-s + (0.404 − 0.233i)19-s + 0.848i·22-s + (0.389 + 0.673i)23-s + (−0.00668 − 0.0115i)26-s + (−0.218 + 0.449i)28-s + 0.166i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.766 - 0.641i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.766 - 0.641i)$
$L(1)$  $\approx$  $0.1387751541$
$L(\frac12)$  $\approx$  $0.1387751541$
$L(\frac{3}{2})$   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 \( 1 + (-2.63 + 0.189i)T \)
good11 \( 1 + (3.44 + 1.99i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.0681T + 13T^{2} \)
17 \( 1 + (6.34 + 3.66i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.76 + 1.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.898iT - 29T^{2} \)
31 \( 1 + (4.18 + 2.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.52 - 2.03i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.68T + 41T^{2} \)
43 \( 1 - 0.964iT - 43T^{2} \)
47 \( 1 + (1.43 - 0.830i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.61 - 11.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.32 - 9.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.51 + 3.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.23 + 5.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.93iT - 71T^{2} \)
73 \( 1 + (5.82 - 10.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.77 - 15.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 + (0.913 + 1.58i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−8.332356067450404587230774362672, −7.59473421475479303520270086300, −7.01193892079372281291650690493, −5.77382422204865904865808342127, −4.99226144364956942486852076942, −4.37314444019068317810661614550, −3.18632256833160326565675341360, −2.42815388847553360499543314273, −1.39589328307325554929370418436, −0.04697120524042884131429270921, 1.61802173360731140562121077526, 2.40428609281301037885508827510, 3.84207035168136673172734155855, 4.81539518245941789874125620495, 5.20367176380586134269775683530, 6.23351308514720574883471462293, 6.99966811456239666341151337140, 7.69703838884630418186534675080, 8.381594521162556864466191939736, 8.856494269705602525619952039104

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