Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.535 - 0.844i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.690 + 2.12i)5-s + (0.809 + 0.587i)6-s + 0.381·7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−1.80 + 1.31i)10-s + (0.427 + 1.31i)11-s + (−0.309 + 0.951i)12-s + (0.763 − 2.35i)13-s + (0.118 + 0.363i)14-s + (1.80 + 1.31i)15-s + (0.309 − 0.951i)16-s + (2.61 + 1.90i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.309 + 0.951i)5-s + (0.330 + 0.239i)6-s + 0.144·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.572 + 0.415i)10-s + (0.128 + 0.396i)11-s + (−0.0892 + 0.274i)12-s + (0.211 − 0.652i)13-s + (0.0315 + 0.0970i)14-s + (0.467 + 0.339i)15-s + (0.0772 − 0.237i)16-s + (0.634 + 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.535 - 0.844i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.535 - 0.844i)$
$L(1)$  $\approx$  $1.23320 + 0.677961i$
$L(\frac12)$  $\approx$  $1.23320 + 0.677961i$
$L(\frac{3}{2})$   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 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.690 - 2.12i)T \)
good7 \( 1 - 0.381T + 7T^{2} \)
11 \( 1 + (-0.427 - 1.31i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.763 + 2.35i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.61 - 1.90i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.23 + 4.53i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.38 + 4.25i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.381 + 0.277i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.54 + 2.57i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.47 + 7.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.38 - 7.33i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + (-9.47 + 6.88i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.35 - 5.34i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.427 - 1.31i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.23 - 6.88i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.47 - 6.15i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (11.7 - 8.50i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.85 - 11.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.73 + 1.98i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.16 + 5.20i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-3.85 - 11.8i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.54 + 3.30i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.26121827704303218769370158445, −12.56900383217744894988950868198, −11.09031684263398783549593562596, −10.08189558369387516896171485341, −8.829523272565290399427249198699, −7.76082225247954562434367029336, −6.79730757470107264759183094159, −5.82372521720221727346789161515, −4.09191407233855373180184861887, −2.54035898568887869145233121268, 1.77705325616513407938804107169, 3.64375192631268278900769211989, 4.78863267738919922204329203112, 6.03746520332227722150304872891, 7.977295075985077516107503227531, 8.937730417884018014575819450147, 9.734748605561883321750643071433, 10.82748300933680381712565068059, 11.98215083339237061554415778453, 12.80564295324049036716903454592

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