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Properties

Label 2-45e2-45.22-c0-0-3
Degree $2$
Conductor $2025$
Sign $0.913 + 0.407i$
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 + (0.448 − 1.67i)7-s + (0.499 + 0.866i)16-s − i·19-s + (1.22 − 1.22i)28-s + (−0.5 + 0.866i)31-s + (−1.22 − 1.22i)37-s + (1.67 + 0.448i)43-s + (−1.73 − 1.00i)49-s + (0.5 + 0.866i)61-s + 0.999i·64-s + (−1.22 + 1.22i)73-s + (0.5 − 0.866i)76-s + (0.866 − 0.5i)79-s + (−0.448 + 1.67i)97-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)4-s + (0.448 − 1.67i)7-s + (0.499 + 0.866i)16-s − i·19-s + (1.22 − 1.22i)28-s + (−0.5 + 0.866i)31-s + (−1.22 − 1.22i)37-s + (1.67 + 0.448i)43-s + (−1.73 − 1.00i)49-s + (0.5 + 0.866i)61-s + 0.999i·64-s + (−1.22 + 1.22i)73-s + (0.5 − 0.866i)76-s + (0.866 − 0.5i)79-s + (−0.448 + 1.67i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.913 + 0.407i$
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: \((2,\ 2025,\ (\ :0),\ 0.913 + 0.407i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.489608030\)
\(L(\frac12)\) \(\approx\) \(1.489608030\)
\(L(1)\) 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 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.216737795057544006435115074712, −8.367322310353833953643150619885, −7.35569274920675792331835541989, −7.27036405915345576972260643325, −6.36842334401683276731892029151, −5.21925586192394950227391090060, −4.20570484197394836533359331899, −3.54077799871259287139537224310, −2.41278222484661717640736737468, −1.18813933527949742346335566926, 1.65268987625112676924360223526, 2.36095229997074058776417923842, 3.34459199736081646583744935230, 4.74299496499213317074705702801, 5.69414854396773770924209541829, 5.95079706104546219642262827804, 6.98842636399451548155082134968, 7.88387099028459176316292914425, 8.590495834147977730050197934345, 9.396760361626715343265024892610

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