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Properties

Label 2-45e2-45.43-c0-0-3
Degree $2$
Conductor $2025$
Sign $0.784 + 0.619i$
Analytic cond. $1.01060$
Root an. cond. $1.00528$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)4-s + (0.499 − 0.866i)16-s − 2i·19-s + (1 + 1.73i)31-s + (0.866 − 0.5i)49-s + (−1 + 1.73i)61-s − 0.999i·64-s + (−1 − 1.73i)76-s + (−1.73 − i)79-s + 2i·109-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)4-s + (0.499 − 0.866i)16-s − 2i·19-s + (1 + 1.73i)31-s + (0.866 − 0.5i)49-s + (−1 + 1.73i)61-s − 0.999i·64-s + (−1 − 1.73i)76-s + (−1.73 − i)79-s + 2i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.784 + 0.619i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (1918, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.784 + 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.433652916\)
\(L(\frac12)\) \(\approx\) \(1.433652916\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.193674334572395517461523906129, −8.596370371596189057681157171906, −7.44402205666228789149031157039, −6.92073902586756470281856170594, −6.20443779759909999053790832833, −5.26037172893445550225758803054, −4.53300959392719772765972151807, −3.14115614302124186429874687079, −2.42428663108645453616896082467, −1.13907241396075989866593339015, 1.57802667734058648196089903246, 2.58481008580532808032742575831, 3.57769221442108024107340462908, 4.35297343121870131940329490653, 5.72400779242350861868074230139, 6.19647346026414920365475411152, 7.15207478590322281747345360878, 7.921260647398529154365385980532, 8.341302863697703287582380983643, 9.523590156562504172737593709961

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