Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.382 - 0.923i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + (0.258 + 0.965i)3-s + 1.00i·4-s + (−0.500 + 0.866i)6-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s − 1.73i·11-s + (−0.965 + 0.258i)12-s − 1.00·16-s + (0.707 + 0.707i)17-s + (−0.965 − 0.258i)18-s i·19-s + (1.22 − 1.22i)22-s + (−0.866 − 0.500i)24-s + (−0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.258 + 0.965i)3-s + 1.00i·4-s + (−0.500 + 0.866i)6-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s − 1.73i·11-s + (−0.965 + 0.258i)12-s − 1.00·16-s + (0.707 + 0.707i)17-s + (−0.965 − 0.258i)18-s i·19-s + (1.22 − 1.22i)22-s + (−0.866 − 0.500i)24-s + (−0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.382 - 0.923i$
motivic weight  =  \(0\)
character  :  $\chi_{600} (443, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :0),\ -0.382 - 0.923i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.333103324\)
\(L(\frac12)\)  \(\approx\)  \(1.333103324\)
\(L(1)\)   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 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−11.23241101596658626293564040405, −10.35062352955300220715088587050, −9.126728597007495566547052058947, −8.500138845903439369586263444893, −7.73994378278289791267014053211, −6.34775508616017304841428482798, −5.63644179244436880119532662276, −4.69713918049670469401304889610, −3.60742329880847449449973536253, −2.88852821851012515310168268816, 1.55063002561232208260280616514, 2.55498299986526963949050251362, 3.79227514851019030460906859615, 4.99548761884097552971721869854, 5.97847321503364693208130348955, 7.02573556671341992045434765408, 7.71255398884353962794670524358, 9.071166489096610887728610664278, 9.832973523586732183179232294526, 10.69705981808287142087854012570

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