Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $-0.169 - 0.985i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (−1.73 + 1.40i)5-s + (−0.809 − 0.587i)6-s + 2.61i·7-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (1.21 − 1.87i)10-s + (−0.0883 − 0.271i)11-s + (0.951 + 0.309i)12-s + (2.66 + 0.866i)13-s + (−0.809 − 2.49i)14-s + (−2.16 − 0.575i)15-s + (0.309 − 0.951i)16-s + (−3.81 + 5.25i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 − 0.293i)4-s + (−0.776 + 0.630i)5-s + (−0.330 − 0.239i)6-s + 0.990i·7-s + (−0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s + (0.384 − 0.593i)10-s + (−0.0266 − 0.0819i)11-s + (0.274 + 0.0892i)12-s + (0.739 + 0.240i)13-s + (−0.216 − 0.665i)14-s + (−0.557 − 0.148i)15-s + (0.0772 − 0.237i)16-s + (−0.925 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.169 - 0.985i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ -0.169 - 0.985i)$
$L(1)$  $\approx$  $0.505809 + 0.600417i$
$L(\frac12)$  $\approx$  $0.505809 + 0.600417i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + (1.73 - 1.40i)T \)
good7 \( 1 - 2.61iT - 7T^{2} \)
11 \( 1 + (0.0883 + 0.271i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.66 - 0.866i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.81 - 5.25i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.358 + 0.260i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-6.20 + 2.01i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.87 + 4.26i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.93 + 2.13i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.31 - 1.40i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.64 + 5.06i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.05iT - 43T^{2} \)
47 \( 1 + (5.28 + 7.27i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-8.17 - 11.2i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.15 + 3.56i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.02 - 12.3i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (6.01 - 8.27i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-3.89 + 2.83i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (11.2 - 3.64i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.32 - 4.59i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.03 + 4.17i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.90 - 5.86i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.20 + 4.40i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.35188512599796550892780852598, −12.01029761189919857012338696912, −11.07377366416786681656494597043, −10.31638901208697139856146324164, −8.823589695489686938720050280432, −8.471470543228727829942396374743, −7.04390069489143081929714223020, −5.89773758838249040776470545442, −4.12374210979952332353847118232, −2.56795626813200928523395999178, 1.00099931379038721721612524985, 3.22704791489889225024808475536, 4.69274515583476323295380974570, 6.75674526189701164704135471915, 7.58572062926356109326642067581, 8.562557258359278271453072588227, 9.452348303912941735645772932966, 10.88424557537417923869743637086, 11.54250186649414845019717782136, 12.81683256130560208666996543487

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