Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} $
Sign $0.221 - 0.975i$
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 + (2.03 − 0.917i)5-s + (0.309 − 0.951i)6-s + 4.80i·7-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (−0.456 + 2.18i)10-s + (0.714 + 0.518i)11-s + (0.587 + 0.809i)12-s + (1.66 + 2.28i)13-s + (−3.88 − 2.82i)14-s + (−1.65 + 1.50i)15-s + (−0.809 + 0.587i)16-s + (−1.57 − 0.512i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.549 + 0.178i)3-s + (−0.154 − 0.475i)4-s + (0.912 − 0.410i)5-s + (0.126 − 0.388i)6-s + 1.81i·7-s + (0.336 + 0.109i)8-s + (0.269 − 0.195i)9-s + (−0.144 + 0.692i)10-s + (0.215 + 0.156i)11-s + (0.169 + 0.233i)12-s + (0.460 + 0.633i)13-s + (−1.03 − 0.755i)14-s + (−0.427 + 0.387i)15-s + (−0.202 + 0.146i)16-s + (−0.382 − 0.124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.221 - 0.975i$
motivic weight  =  \(1\)
character  :  $\chi_{150} (109, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 150,\ (\ :1/2),\ 0.221 - 0.975i)$
$L(1)$  $\approx$  $0.684296 + 0.546488i$
$L(\frac12)$  $\approx$  $0.684296 + 0.546488i$
$L(\frac{3}{2})$   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 \( 1 + (-2.03 + 0.917i)T \)
good7 \( 1 - 4.80iT - 7T^{2} \)
11 \( 1 + (-0.714 - 0.518i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.66 - 2.28i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.57 + 0.512i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.66 - 5.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.44 + 4.73i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.10 + 3.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.22 + 9.93i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.02 + 1.41i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.40 + 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.27iT - 43T^{2} \)
47 \( 1 + (8.29 - 2.69i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.37 - 1.09i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.37 + 6.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.0697 - 0.0506i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (11.3 + 3.69i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.08 + 3.33i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.24 + 5.84i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.88 - 11.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (12.6 + 4.10i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-15.1 - 10.9i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-17.3 + 5.64i)T + (78.4 - 57.0i)T^{2} \)
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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

−13.14182453488250183004207302168, −12.23832312293567736505971817238, −11.22315859791817778019525790502, −9.845145002999002364443474160857, −9.145024981551667692816311991064, −8.282794610117585009581090768757, −6.35746047367527950596297032892, −5.90441883007124945679377650570, −4.71006397044532192505715661509, −2.08030982605615545982831490121, 1.23334448227241173793524948406, 3.33495566488987431538765481070, 4.91337899669824332755938034696, 6.58296775625480162173867053109, 7.31993191135650874057816157105, 8.862698797747793053735989363724, 10.16621043982084590843709450452, 10.66405445554444785840531750309, 11.42815315736287151260344632351, 13.12269424889476471201229767950

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