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Properties

Label 2-45e2-45.22-c0-0-0
Degree $2$
Conductor $2025$
Sign $-0.929 - 0.370i$
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  + (−0.448 + 1.67i)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 + (0.448 + 1.67i)23-s + (−0.5 + 0.866i)31-s + (−2.59 + 1.5i)34-s + (1.67 + 0.448i)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  + (−0.448 + 1.67i)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 + (0.448 + 1.67i)23-s + (−0.5 + 0.866i)31-s + (−2.59 + 1.5i)34-s + (1.67 + 0.448i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $-0.929 - 0.370i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ -0.929 - 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7917841437\)
\(L(\frac12)\) \(\approx\) \(0.7917841437\)
\(L(1)\) 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 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
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

−9.301988265804572592912333372229, −8.800429725717111628234847774942, −7.894872816927261182084227609854, −7.41851266708255396250063298360, −6.65634200192204850595994274915, −5.76553350246512544902265646387, −5.30586596627721320757874152539, −4.27105207027614702451851778858, −3.15417557356665821861929249629, −1.34072867693316563923049959516, 0.75923996546691384249376715489, 2.00505530340631323964884036226, 2.91790504771444762260053635574, 3.70045061984745150653111003066, 4.62923061854536508681877531381, 5.55502646170678423085201476134, 6.74093396486084067469018485759, 7.80726871342299415625009643829, 8.440550779664476881237491693804, 9.308472815041178705342511533907

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