Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.788 + 0.615i$
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 + (1.86 + 1.23i)5-s + (0.309 − 0.951i)6-s + 2.70i·7-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (2.09 − 0.786i)10-s + (−4.54 − 3.30i)11-s + (−0.587 − 0.809i)12-s + (−2.84 − 3.91i)13-s + (2.19 + 1.59i)14-s + (2.15 + 0.593i)15-s + (−0.809 + 0.587i)16-s + (0.994 + 0.323i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (0.549 − 0.178i)3-s + (−0.154 − 0.475i)4-s + (0.834 + 0.550i)5-s + (0.126 − 0.388i)6-s + 1.02i·7-s + (−0.336 − 0.109i)8-s + (0.269 − 0.195i)9-s + (0.661 − 0.248i)10-s + (−1.37 − 0.996i)11-s + (−0.169 − 0.233i)12-s + (−0.789 − 1.08i)13-s + (0.585 + 0.425i)14-s + (0.556 + 0.153i)15-s + (−0.202 + 0.146i)16-s + (0.241 + 0.0783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.788 + 0.615i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (109, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.788 + 0.615i)$
$L(1)$  $\approx$  $1.52551 - 0.525218i$
$L(\frac12)$  $\approx$  $1.52551 - 0.525218i$
$L(\frac{3}{2})$   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 \( 1 + (-1.86 - 1.23i)T \)
good7 \( 1 - 2.70iT - 7T^{2} \)
11 \( 1 + (4.54 + 3.30i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.84 + 3.91i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.994 - 0.323i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.59 - 7.97i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.11 + 4.28i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.29 + 3.98i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.72 - 5.30i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.75 - 2.42i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.27 + 0.927i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.29iT - 43T^{2} \)
47 \( 1 + (-2.92 + 0.949i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.87 + 1.90i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (11.4 - 8.33i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.218 + 0.159i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-6.00 - 1.94i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.40 + 4.31i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.79 + 6.60i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.85 + 11.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.127 - 0.0415i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-9.53 - 6.93i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.51 + 0.815i)T + (78.4 - 57.0i)T^{2} \)
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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

−12.88695961900462816433901218163, −12.22149115396931016591079474486, −10.63662698156745087546743999760, −10.17872685223726168796089363820, −8.844733151914900358329161221760, −7.81440162727950482049853341657, −6.04717352844010059578141010441, −5.32404186214776755132007573485, −3.15395405445264713373292220980, −2.32219329465399738568902724861, 2.42022892764829301230525190014, 4.40163314503455639244310349107, 5.15144116589814311900977525527, 6.91354326901691711319421766417, 7.61943890849607678080057446756, 9.091763664221481667283389961702, 9.805108586721541229147584507842, 11.01065644833086812424011744491, 12.68214395471470829740089472490, 13.25565817447270080944733859589

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