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Properties

Label 2-45e2-45.13-c0-0-1
Degree $2$
Conductor $2025$
Sign $0.746 - 0.665i$
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  + (−1.67 − 0.448i)2-s + (1.73 + 1.00i)4-s + (−1.22 − 1.22i)8-s + (0.500 + 0.866i)16-s + (−1.22 + 1.22i)17-s + i·19-s + (1.67 − 0.448i)23-s + (−0.5 + 0.866i)31-s + (2.59 − 1.5i)34-s + (0.448 − 1.67i)38-s − 3·46-s + (−0.866 − 0.5i)49-s + (1.22 + 1.22i)53-s + (0.5 + 0.866i)61-s + (1.22 − 1.22i)62-s + ⋯
L(s)  = 1  + (−1.67 − 0.448i)2-s + (1.73 + 1.00i)4-s + (−1.22 − 1.22i)8-s + (0.500 + 0.866i)16-s + (−1.22 + 1.22i)17-s + i·19-s + (1.67 − 0.448i)23-s + (−0.5 + 0.866i)31-s + (2.59 − 1.5i)34-s + (0.448 − 1.67i)38-s − 3·46-s + (−0.866 − 0.5i)49-s + (1.22 + 1.22i)53-s + (0.5 + 0.866i)61-s + (1.22 − 1.22i)62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4447831628\)
\(L(\frac12)\) \(\approx\) \(0.4447831628\)
\(L(1)\) 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 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.353320904579111808917210913654, −8.600773849645654657451327081785, −8.311075395163406286057200346574, −7.20201627374106224229446211469, −6.72589643732588647164198879011, −5.65461672745109609268173058429, −4.38377407365599445020913554225, −3.25094727054297624521342535808, −2.21160917173631827074362482782, −1.25371859015343468351922359598, 0.60404885934576777908142592637, 2.00516071898604900044804170382, 2.98814286099046736467860303053, 4.53698281064382754248480463753, 5.44991887435929081662220757250, 6.61146940536348479169952910610, 7.03429044324291931548068294799, 7.70481126101819833796083591431, 8.696576163808604614556366669943, 9.138196957414982504811386665891

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