Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 5^{2} $
Sign $-0.881 - 0.472i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.881 - 0.472i$
motivic weight  =  \(1\)
character  :  $\chi_{2400} (1199, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2400,\ (\ :1/2),\ -0.881 - 0.472i)\)
\(L(1)\)  \(\approx\)  \(0.4216305659\)
\(L(\frac12)\)  \(\approx\)  \(0.4216305659\)
\(L(\frac{3}{2})\)   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

−9.424888659609537304758950988763, −8.607843011815027373766475013817, −7.80943905238900101324431945307, −6.61963242881428181161187029389, −6.29271407655236427610744979862, −5.28303009069981396384737105043, −4.61765614760641143620334831032, −3.73192359101095674866413692047, −2.76257739523020073045330005363, −1.20863639602178718507715692453, 0.17013275168272142208483974471, 1.64630922937879904292858501678, 2.47819439129769698588424574198, 3.94142737416113330815536991514, 4.78530692609946303248675764718, 5.49856286604428260017892730429, 6.41804476956842995249193085505, 7.06276705956025717116034359040, 7.62382553203900393763334923509, 8.635220474065456136030836545036

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