Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 5^{2} $
Sign $0.151 - 0.988i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + i)3-s + (1.00 + 2.82i)9-s + 2.82i·11-s + 5.65·17-s + 2·19-s + (−1.41 + 5.00i)27-s + (−2.82 + 4.00i)33-s − 11.3i·41-s + 10i·43-s − 7·49-s + (8.00 + 5.65i)51-s + (2.82 + 2i)57-s + 14.1i·59-s + 14i·67-s + 2i·73-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)3-s + (0.333 + 0.942i)9-s + 0.852i·11-s + 1.37·17-s + 0.458·19-s + (−0.272 + 0.962i)27-s + (−0.492 + 0.696i)33-s − 1.76i·41-s + 1.52i·43-s − 49-s + (1.12 + 0.792i)51-s + (0.374 + 0.264i)57-s + 1.84i·59-s + 1.71i·67-s + 0.234i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.151 - 0.988i$
motivic weight  =  \(1\)
character  :  $\chi_{2400} (1199, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2400,\ (\ :1/2),\ 0.151 - 0.988i)\)
\(L(1)\)  \(\approx\)  \(2.457443027\)
\(L(\frac12)\)  \(\approx\)  \(2.457443027\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 + (-1.41 - i)T \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.178925302437678571678436151792, −8.422598607456698994882595113445, −7.60418155854909740728358632896, −7.15325432972950077837599709832, −5.87252255427914855921912953853, −5.07377034194210019090358183916, −4.25428065289298049354292789830, −3.40654482431188597024711804757, −2.54982160730103367601115448735, −1.41288938563407289484571452043, 0.798144473446670988925298395105, 1.88440510793638165811270605177, 3.14716312487960982634276622200, 3.51019217127761944155256999009, 4.81486791749538356176553442683, 5.81113352523631552509028784381, 6.51655864340830537046825341026, 7.44174023517139402366067105455, 8.036636627211214042951399462274, 8.624021206907247523595393642964

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