Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.728 - 0.684i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (1.80 − 1.31i)5-s + (−0.309 − 0.951i)6-s + 2.61·7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.690 + 2.12i)10-s + (−2.92 + 2.12i)11-s + (0.809 + 0.587i)12-s + (5.23 + 3.80i)13-s + (−2.11 + 1.53i)14-s + (0.690 + 2.12i)15-s + (−0.809 − 0.587i)16-s + (0.381 + 1.17i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (0.809 − 0.587i)5-s + (−0.126 − 0.388i)6-s + 0.989·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.218 + 0.672i)10-s + (−0.882 + 0.641i)11-s + (0.233 + 0.169i)12-s + (1.45 + 1.05i)13-s + (−0.566 + 0.411i)14-s + (0.178 + 0.549i)15-s + (−0.202 − 0.146i)16-s + (0.0926 + 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.728 - 0.684i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (61, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.728 - 0.684i)$
$L(1)$  $\approx$  $0.904965 + 0.358301i$
$L(\frac12)$  $\approx$  $0.904965 + 0.358301i$
$L(\frac{3}{2})$   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 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-1.80 + 1.31i)T \)
good7 \( 1 - 2.61T + 7T^{2} \)
11 \( 1 + (2.92 - 2.12i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-5.23 - 3.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.381 - 1.17i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.76 + 5.42i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.61 - 2.62i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.61 + 8.05i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.04 - 6.29i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.47 + 4.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.61 + 3.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.70T + 43T^{2} \)
47 \( 1 + (-0.527 + 1.62i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.645 - 1.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.92 - 2.12i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.472 + 1.45i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.70 + 5.25i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.85 - 2.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.73 - 5.34i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.663 - 2.04i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.85 - 2.07i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.04 - 3.21i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.42435542664339969136864478086, −11.91417337427523921243828018804, −10.88071608785578157117876434923, −10.02694587565254672946413257248, −8.920591426148380215598855230777, −8.235068105895055738684134390160, −6.65649491396160214640373799445, −5.44651566070203651198041074908, −4.46643273137775871687519530667, −1.84506009937836413901459379786, 1.60803581944367060453600804459, 3.17409085559959444532726007576, 5.40866165873349448291739280325, 6.44167096348144035519339700530, 7.987934348601374212507366620492, 8.466324681271482704732631883474, 10.23783206112872468197290733539, 10.71680349504585376026666136175, 11.67929241500716773058219857480, 12.94291777957140005444650409487

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