Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 5^{2} $
Sign $-0.935 - 0.353i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3.62i·7-s + 9-s + 6.20i·11-s + 0.578·13-s + 1.42i·17-s + 5.62i·19-s − 3.62i·21-s − 5.62i·23-s − 27-s + 2i·29-s + 2.57·31-s − 6.20i·33-s − 7.83·37-s − 0.578·39-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.37i·7-s + 0.333·9-s + 1.87i·11-s + 0.160·13-s + 0.344i·17-s + 1.29i·19-s − 0.791i·21-s − 1.17i·23-s − 0.192·27-s + 0.371i·29-s + 0.463·31-s − 1.08i·33-s − 1.28·37-s − 0.0926·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.935 - 0.353i$
motivic weight  =  \(1\)
character  :  $\chi_{2400} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2400,\ (\ :1/2),\ -0.935 - 0.353i)\)
\(L(1)\)  \(\approx\)  \(0.9539871803\)
\(L(\frac12)\)  \(\approx\)  \(0.9539871803\)
\(L(\frac{3}{2})\)   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 \( 1 + T \)
5 \( 1 \)
good7 \( 1 - 3.62iT - 7T^{2} \)
11 \( 1 - 6.20iT - 11T^{2} \)
13 \( 1 - 0.578T + 13T^{2} \)
17 \( 1 - 1.42iT - 17T^{2} \)
19 \( 1 - 5.62iT - 19T^{2} \)
23 \( 1 + 5.62iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 2.57T + 31T^{2} \)
37 \( 1 + 7.83T + 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 + 7.25T + 43T^{2} \)
47 \( 1 + 6.78iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 2.20iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 5.42T + 79T^{2} \)
83 \( 1 - 3.25T + 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 - 4.84iT - 97T^{2} \)
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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

−9.317288982282382623501838362649, −8.565817161110790381000909798307, −7.79117738234769431505404394834, −6.82612651431872555189681327329, −6.22285651218383833719852143859, −5.32443562186763852234967095773, −4.73816400500375059151741526361, −3.72293630333557068677002239686, −2.38123385107897960401535928205, −1.66917313509841299267028713508, 0.37447667192830236477776389283, 1.24259243143419065276927415183, 2.96908126547384873592848117038, 3.73551998677796310133921823607, 4.64240621082944485346952258537, 5.52900417118742267813558465146, 6.29724149377622486041225536729, 7.07069383686250477454671984017, 7.72549313057596249835727000512, 8.635705569009749909433863647113

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