Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.435 + 0.900i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−1.53 − 1.62i)5-s + (0.309 − 0.951i)6-s − 4.63i·7-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (2.21 − 0.292i)10-s + (2.05 + 1.49i)11-s + (0.587 + 0.809i)12-s + (0.0846 + 0.116i)13-s + (3.74 + 2.72i)14-s + (1.96 + 1.06i)15-s + (−0.809 + 0.587i)16-s + (−7.12 − 2.31i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.549 + 0.178i)3-s + (−0.154 − 0.475i)4-s + (−0.688 − 0.725i)5-s + (0.126 − 0.388i)6-s − 1.75i·7-s + (0.336 + 0.109i)8-s + (0.269 − 0.195i)9-s + (0.701 − 0.0923i)10-s + (0.619 + 0.450i)11-s + (0.169 + 0.233i)12-s + (0.0234 + 0.0323i)13-s + (1.00 + 0.727i)14-s + (0.507 + 0.275i)15-s + (−0.202 + 0.146i)16-s + (−1.72 − 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.435 + 0.900i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (109, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.435 + 0.900i)$
$L(1)$  $\approx$  $0.495005 - 0.310442i$
$L(\frac12)$  $\approx$  $0.495005 - 0.310442i$
$L(\frac{3}{2})$   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 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (1.53 + 1.62i)T \)
good7 \( 1 + 4.63iT - 7T^{2} \)
11 \( 1 + (-2.05 - 1.49i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.0846 - 0.116i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (7.12 + 2.31i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.08 + 6.41i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.985 + 1.35i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.696 - 2.14i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.310 + 0.954i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.0523 + 0.0719i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.48 - 1.80i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.02iT - 43T^{2} \)
47 \( 1 + (-10.3 + 3.36i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.72 + 1.53i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.25 + 3.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-11.0 - 8.05i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-7.27 - 2.36i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.17 + 9.76i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.15 + 1.59i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.230 + 0.710i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.95 + 0.958i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-0.593 - 0.431i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (8.67 - 2.81i)T + (78.4 - 57.0i)T^{2} \)
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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

−13.02336679306510910988181914441, −11.54297377451137033371077814924, −10.89260124656019385867119173191, −9.669433209087033264327564211641, −8.698791561969935880056756805480, −7.26015038316085706809114721532, −6.80094948883076233108958526555, −4.84727930444327479432444390268, −4.17502788732113545120585188542, −0.72458716210564977758539970976, 2.28465551055071685817340172735, 3.85884944433142228619913926406, 5.67302619402657758738404970952, 6.75756041820781196392244105806, 8.219900250433420488334197887326, 9.025224089794356986421227455695, 10.37265229373609522697633105060, 11.42346665004226686849983025122, 11.89081662487521644905381481007, 12.74082230435028126132585804961

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