Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $-0.699 + 0.714i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.891 − 0.453i)2-s + (−0.322 − 1.70i)3-s + (0.587 + 0.809i)4-s + (−0.0831 − 2.23i)5-s + (−0.485 + 1.66i)6-s + (0.0556 + 0.0556i)7-s + (−0.156 − 0.987i)8-s + (−2.79 + 1.09i)9-s + (−0.940 + 2.02i)10-s + (−1.04 + 0.340i)11-s + (1.18 − 1.26i)12-s + (−2.31 − 4.54i)13-s + (−0.0243 − 0.0748i)14-s + (−3.77 + 0.861i)15-s + (−0.309 + 0.951i)16-s + (3.10 − 0.491i)17-s + ⋯
L(s)  = 1  + (−0.630 − 0.321i)2-s + (−0.186 − 0.982i)3-s + (0.293 + 0.404i)4-s + (−0.0371 − 0.999i)5-s + (−0.198 + 0.678i)6-s + (0.0210 + 0.0210i)7-s + (−0.0553 − 0.349i)8-s + (−0.930 + 0.365i)9-s + (−0.297 + 0.641i)10-s + (−0.315 + 0.102i)11-s + (0.342 − 0.364i)12-s + (−0.641 − 1.25i)13-s + (−0.00649 − 0.0200i)14-s + (−0.974 + 0.222i)15-s + (−0.0772 + 0.237i)16-s + (0.752 − 0.119i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.699 + 0.714i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ -0.699 + 0.714i)$
$L(1)$  $\approx$  $0.269657 - 0.641732i$
$L(\frac12)$  $\approx$  $0.269657 - 0.641732i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (0.891 + 0.453i)T \)
3 \( 1 + (0.322 + 1.70i)T \)
5 \( 1 + (0.0831 + 2.23i)T \)
good7 \( 1 + (-0.0556 - 0.0556i)T + 7iT^{2} \)
11 \( 1 + (1.04 - 0.340i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.31 + 4.54i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-3.10 + 0.491i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (0.824 - 1.13i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.13 + 2.22i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-5.66 + 4.11i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-7.72 - 5.61i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-9.23 + 4.70i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-3.55 - 1.15i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.00 - 1.00i)T - 43iT^{2} \)
47 \( 1 + (1.98 - 12.5i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (6.70 + 1.06i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-2.03 + 6.26i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.23 - 3.79i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.04 + 6.57i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (7.51 + 10.3i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.18 - 0.602i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-5.85 - 8.05i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.137 + 0.871i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-1.01 - 3.11i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (10.2 + 1.62i)T + (92.2 + 29.9i)T^{2} \)
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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

−12.52083641953973657296437105808, −11.90365442531089614120176829752, −10.61134594379741979395064410437, −9.534721323742546718273289114631, −8.161568046784006854802113441678, −7.81760045807464976227036302199, −6.23099428388710134336528103501, −4.94469792095970612420716043400, −2.71671810405974884994271828035, −0.887741335939681819068339936386, 2.81488393560064313043482143393, 4.47579571289746162711663817432, 5.94955947085463841935864698689, 7.01669482648809182821527441367, 8.278819696073703186805943905657, 9.564718073487864982083422274189, 10.15425732365626212574693645266, 11.18783605813711700834153698987, 11.90102896494315344334862271241, 13.79807596108987755090890888966

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