Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.244 - 0.969i$
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 + (−1.36 + 1.76i)5-s + (0.309 + 0.951i)6-s + 0.533i·7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−2.23 − 0.0655i)10-s + (1.16 − 0.843i)11-s + (−0.587 + 0.809i)12-s + (3.86 − 5.31i)13-s + (−0.431 + 0.313i)14-s + (−1.84 + 1.26i)15-s + (−0.809 − 0.587i)16-s + (−0.911 + 0.296i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (−0.611 + 0.791i)5-s + (0.126 + 0.388i)6-s + 0.201i·7-s + (−0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s + (−0.706 − 0.0207i)10-s + (0.349 − 0.254i)11-s + (−0.169 + 0.233i)12-s + (1.07 − 1.47i)13-s + (−0.115 + 0.0838i)14-s + (−0.476 + 0.325i)15-s + (−0.202 − 0.146i)16-s + (−0.221 + 0.0718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.244 - 0.969i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (139, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.244 - 0.969i)$
$L(1)$  $\approx$  $1.15081 + 0.896513i$
$L(\frac12)$  $\approx$  $1.15081 + 0.896513i$
$L(\frac{3}{2})$   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 \( 1 + (1.36 - 1.76i)T \)
good7 \( 1 - 0.533iT - 7T^{2} \)
11 \( 1 + (-1.16 + 0.843i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.86 + 5.31i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.911 - 0.296i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.0657 + 0.202i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.21 + 3.04i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.91 - 5.89i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.722 + 2.22i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.38 - 3.28i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.42 - 4.66i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 + (9.65 + 3.13i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.07 + 0.999i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.08 - 4.42i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (10.1 - 7.38i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (6.57 - 2.13i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.12 + 9.62i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.21 - 11.3i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.79 - 14.7i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (15.5 - 5.04i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-4.54 + 3.30i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.29 - 1.71i)T + (78.4 + 57.0i)T^{2} \)
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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

−13.41922189338652822801971847973, −12.42985475748562595072005413914, −11.19341946321714570343154124979, −10.29015414977629975329932560060, −8.734185803863746523745961042795, −8.002350724092422712452770297951, −6.85586525759826438001398377917, −5.67669714469046720619578466837, −3.99952350241038515715744764415, −3.00502029327306916201184719055, 1.65590536246921835616227913356, 3.69778008129017253310306431079, 4.52611627518591872764752929848, 6.24403500286325359947602502488, 7.64540504024528231238360153524, 8.860055374591655973469074351657, 9.554435409705503735761707785487, 11.13152518035631910081461671384, 11.83497007531555773495267874534, 12.84243963383822985083518028628

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