Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $-0.212 - 0.977i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.891 + 0.453i)2-s + (−1.37 + 1.05i)3-s + (0.587 + 0.809i)4-s + (−1.04 + 1.97i)5-s + (−1.70 + 0.313i)6-s + (0.462 + 0.462i)7-s + (0.156 + 0.987i)8-s + (0.784 − 2.89i)9-s + (−1.82 + 1.28i)10-s + (−2.73 + 0.888i)11-s + (−1.66 − 0.494i)12-s + (1.97 + 3.86i)13-s + (0.202 + 0.621i)14-s + (−0.641 − 3.81i)15-s + (−0.309 + 0.951i)16-s + (5.75 − 0.911i)17-s + ⋯
L(s)  = 1  + (0.630 + 0.321i)2-s + (−0.794 + 0.607i)3-s + (0.293 + 0.404i)4-s + (−0.467 + 0.883i)5-s + (−0.695 + 0.127i)6-s + (0.174 + 0.174i)7-s + (0.0553 + 0.349i)8-s + (0.261 − 0.965i)9-s + (−0.578 + 0.406i)10-s + (−0.824 + 0.267i)11-s + (−0.479 − 0.142i)12-s + (0.546 + 1.07i)13-s + (0.0539 + 0.166i)14-s + (−0.165 − 0.986i)15-s + (−0.0772 + 0.237i)16-s + (1.39 − 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.212 - 0.977i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ -0.212 - 0.977i)$
$L(1)$  $\approx$  $0.725142 + 0.899315i$
$L(\frac12)$  $\approx$  $0.725142 + 0.899315i$
$L(\frac{3}{2})$   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 \( 1 + (-0.891 - 0.453i)T \)
3 \( 1 + (1.37 - 1.05i)T \)
5 \( 1 + (1.04 - 1.97i)T \)
good7 \( 1 + (-0.462 - 0.462i)T + 7iT^{2} \)
11 \( 1 + (2.73 - 0.888i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.97 - 3.86i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-5.75 + 0.911i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-4.28 + 5.89i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.316 + 0.622i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-2.60 + 1.89i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.82 + 3.50i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.93 - 0.988i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-6.34 - 2.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (5.08 - 5.08i)T - 43iT^{2} \)
47 \( 1 + (0.474 - 2.99i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-2.23 - 0.353i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (2.66 - 8.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.77 - 8.55i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.22 + 14.0i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (7.15 + 9.84i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.498 - 0.254i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-6.00 - 8.25i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.742 + 4.68i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (1.78 + 5.49i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.26 - 0.833i)T + (92.2 + 29.9i)T^{2} \)
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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

−13.38266210382368270420713169821, −12.01723051866921715171155560632, −11.46450600693531104496420334671, −10.54677827648746695885471970925, −9.374817315283107429471912084036, −7.70788312338096007230901557578, −6.74055420285992137469560405603, −5.57884745750058843778570604333, −4.44381569655266412161972951696, −3.10807683729975728615594109655, 1.19148255874697230776117067122, 3.52798553401659911512543196569, 5.24235495659356360620187352331, 5.68595363049078333295539595573, 7.48893963352225065078903368866, 8.217074764346506085161274625657, 10.09106821597752630992854368201, 10.92689824089302746884274231239, 12.06586124632489045949366392394, 12.56895933627508432661524366872

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