↑

Properties

Label 2-45e2-9.5-c0-0-1
Degree $2$
Conductor $2025$
Sign $0.984 - 0.173i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + i·8-s + (0.5 + 0.866i)16-s + i·17-s + 19-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)34-s + (0.866 − 0.5i)38-s + 0.999·46-s + (−1.73 + i)47-s + (0.5 − 0.866i)49-s − i·53-s + (0.5 + 0.866i)61-s − 0.999i·62-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + i·8-s + (0.5 + 0.866i)16-s + i·17-s + 19-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)34-s + (0.866 − 0.5i)38-s + 0.999·46-s + (−1.73 + i)47-s + (0.5 − 0.866i)49-s − i·53-s + (0.5 + 0.866i)61-s − 0.999i·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.751394981\)
\(L(\frac12)\) \(\approx\) \(1.751394981\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.420421703033903586561354694263, −8.492453054321139576679348808252, −7.896012935575039874768661860791, −6.95710419615014115956563553175, −5.91177179035889751102159309482, −5.22382278892486172834765675987, −4.37285045112079370579871923867, −3.54517670398687785612810433522, −2.78490642555186432135423112300, −1.59317375478164994171015730582, 1.10275190080177178849316301460, 2.78742674956118604913233033635, 3.61861895121589991387926485869, 4.76106910382948079635209442445, 5.12979085574737022115734054580, 6.07955915592130985422612419225, 6.88104858651300994514661298742, 7.42879372833329085781764846581, 8.516265375041101473954751608399, 9.378421935260178009171174395914

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