Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 5^{2} $
Sign $0.0310 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4.68i·7-s + 9-s − 2.29i·11-s + 4.97·13-s + 2.97i·17-s + 2.68i·19-s − 4.68i·21-s − 2.68i·23-s − 27-s − 2i·29-s + 6.97·31-s + 2.29i·33-s + 4.39·37-s − 4.97·39-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.77i·7-s + 0.333·9-s − 0.691i·11-s + 1.38·13-s + 0.722i·17-s + 0.616i·19-s − 1.02i·21-s − 0.560i·23-s − 0.192·27-s − 0.371i·29-s + 1.25·31-s + 0.399i·33-s + 0.722·37-s − 0.797·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0310 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0310 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.0310 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{2400} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2400,\ (\ :1/2),\ 0.0310 - 0.999i)\)
\(L(1)\)  \(\approx\)  \(1.418250776\)
\(L(\frac12)\)  \(\approx\)  \(1.418250776\)
\(L(\frac{3}{2})\)   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 \( 1 - 4.68iT - 7T^{2} \)
11 \( 1 + 2.29iT - 11T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 - 2.97iT - 17T^{2} \)
19 \( 1 - 2.68iT - 19T^{2} \)
23 \( 1 + 2.68iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 6.97T + 31T^{2} \)
37 \( 1 - 4.39T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 - 9.37T + 43T^{2} \)
47 \( 1 - 7.27iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 1.70iT - 59T^{2} \)
61 \( 1 - 4.58iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 1.02T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 - 3.95iT - 97T^{2} \)
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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

−8.893948204729404010132886326067, −8.554256138720850927108114534234, −7.81714457236639567265712233824, −6.30493582344699550600534132051, −6.15482679441086505568128651586, −5.47464514385573775393390803243, −4.43271736156641206076866062438, −3.39323298630402956298264283835, −2.40317585666911572209884099399, −1.21391242591165334624701638485, 0.60533226567943684770470271457, 1.53049195324949182375990917473, 3.14071082654360387145378940391, 4.09309291493534630291159479639, 4.64050388347581146855068942298, 5.64175729498501102887656793991, 6.70631863133614594610455683004, 7.04057120406156320249661298158, 7.85784637966337449074587605655, 8.747373732451683301699287631232

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