Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.249 - 0.968i$
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 + (−1.47 + 1.67i)5-s + (−0.809 + 0.587i)6-s + 3.23i·7-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (−1.92 + 1.13i)10-s + (1.63 − 5.04i)11-s + (−0.951 + 0.309i)12-s + (5.61 − 1.82i)13-s + (−0.998 + 3.07i)14-s + (−0.486 − 2.18i)15-s + (0.309 + 0.951i)16-s + (−0.602 − 0.828i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (−0.339 + 0.467i)3-s + (0.404 + 0.293i)4-s + (−0.661 + 0.749i)5-s + (−0.330 + 0.239i)6-s + 1.22i·7-s + (0.207 + 0.286i)8-s + (−0.103 − 0.317i)9-s + (−0.608 + 0.359i)10-s + (0.493 − 1.51i)11-s + (−0.274 + 0.0892i)12-s + (1.55 − 0.506i)13-s + (−0.266 + 0.821i)14-s + (−0.125 − 0.563i)15-s + (0.0772 + 0.237i)16-s + (−0.146 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.249 - 0.968i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (79, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.249 - 0.968i)$
$L(1)$  $\approx$  $1.07485 + 0.832867i$
$L(\frac12)$  $\approx$  $1.07485 + 0.832867i$
$L(\frac{3}{2})$   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 \( 1 + (1.47 - 1.67i)T \)
good7 \( 1 - 3.23iT - 7T^{2} \)
11 \( 1 + (-1.63 + 5.04i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-5.61 + 1.82i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.602 + 0.828i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (3.64 - 2.64i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.83 + 0.595i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.210 + 0.153i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.81 + 1.31i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-9.08 + 2.95i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.07 + 9.47i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.30iT - 43T^{2} \)
47 \( 1 + (6.18 - 8.51i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.35 + 1.86i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.00 - 6.17i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.21 - 3.75i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (5.46 + 7.51i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (6.90 + 5.01i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.76 + 1.22i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.8 + 7.85i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.40 - 6.05i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.226 + 0.697i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (10.1 - 13.9i)T + (-29.9 - 92.2i)T^{2} \)
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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

−13.27713052707075033420199128056, −12.04994364184353744132112264127, −11.31471878847110726211021186484, −10.64533072550843214506812965065, −8.883300455334421701880046394330, −8.075689342540756307936498147392, −6.21766553125854739154450357857, −5.89511123421341433252662986132, −4.06123638027918810283498643775, −3.02998996966798272484720143825, 1.43813782509858404915512192485, 3.96647364997502156248679473321, 4.63719878087666217151293325967, 6.39510539922794811930450753657, 7.25824550449524498074808933051, 8.498543516991817398954458142885, 9.981655302245061205960933295999, 11.16013317483254878641620886432, 11.84356988169692893572020763452, 13.00049774864493051143603025520

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