Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.830 + 0.556i$
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 + (1.97 + 1.03i)5-s + (−0.809 − 0.587i)6-s + 0.329i·7-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (2.20 + 0.377i)10-s + (−1.55 − 4.78i)11-s + (−0.951 − 0.309i)12-s + (−0.458 − 0.148i)13-s + (0.101 + 0.313i)14-s + (−0.322 − 2.21i)15-s + (0.309 − 0.951i)16-s + (−3.98 + 5.49i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (−0.339 − 0.467i)3-s + (0.404 − 0.293i)4-s + (0.885 + 0.465i)5-s + (−0.330 − 0.239i)6-s + 0.124i·7-s + (0.207 − 0.286i)8-s + (−0.103 + 0.317i)9-s + (0.696 + 0.119i)10-s + (−0.468 − 1.44i)11-s + (−0.274 − 0.0892i)12-s + (−0.127 − 0.0413i)13-s + (0.0271 + 0.0837i)14-s + (−0.0832 − 0.571i)15-s + (0.0772 − 0.237i)16-s + (−0.967 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.830 + 0.556i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.830 + 0.556i)$
$L(1)$  $\approx$  $1.49325 - 0.454014i$
$L(\frac12)$  $\approx$  $1.49325 - 0.454014i$
$L(\frac{3}{2})$   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 \( 1 + (-1.97 - 1.03i)T \)
good7 \( 1 - 0.329iT - 7T^{2} \)
11 \( 1 + (1.55 + 4.78i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.458 + 0.148i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.98 - 5.49i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.40 - 3.19i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (6.18 - 2.00i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.87 - 3.53i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.06 + 0.775i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.741 + 0.241i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.86 + 11.9i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.47iT - 43T^{2} \)
47 \( 1 + (2.57 + 3.54i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.990 - 1.36i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.313 - 0.966i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.29 - 3.98i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.84 - 2.54i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (4.62 - 3.35i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.79 - 0.909i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.86 + 4.98i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.1 + 13.9i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.06 + 3.28i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.58 + 7.69i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.11305728372609223959019713783, −12.00697109483912318684347441468, −10.94999638818489656162468480775, −10.28111268880460392672964373608, −8.785061690283155803530051936442, −7.38457221106242170189901786597, −5.99304371920815135935521687174, −5.61625221165410735487927697390, −3.57941304093049904094490642720, −2.00379005318336433334260222751, 2.41630060558005889507551185396, 4.48155873332314157515554019462, 5.16932412868252355270109046132, 6.45578439608469279532235131853, 7.60971292684151884602697341560, 9.316981533567339933114912359443, 9.937592161785444454364363820112, 11.25231525207314674019467233058, 12.24902699137406613551725087023, 13.19937901609202438511616609890

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