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Properties

Label 2-45e2-9.2-c0-0-3
Degree $2$
Conductor $2025$
Sign $-0.642 + 0.766i$
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  + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s + (−1 − 1.73i)13-s + (−0.499 + 0.866i)16-s − 19-s − 0.999·28-s + (0.5 + 0.866i)31-s − 37-s + (0.5 − 0.866i)43-s + (−0.999 + 1.73i)52-s + (0.5 − 0.866i)61-s + 0.999·64-s + (−1 − 1.73i)67-s − 73-s + (0.5 + 0.866i)76-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s + (−1 − 1.73i)13-s + (−0.499 + 0.866i)16-s − 19-s − 0.999·28-s + (0.5 + 0.866i)31-s − 37-s + (0.5 − 0.866i)43-s + (−0.999 + 1.73i)52-s + (0.5 − 0.866i)61-s + 0.999·64-s + (−1 − 1.73i)67-s − 73-s + (0.5 + 0.866i)76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8066796392\)
\(L(\frac12)\) \(\approx\) \(0.8066796392\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.069782732999110643444712638795, −8.269798945479221893721929886097, −7.56084437161882902082301447866, −6.69627011726260919230135636783, −5.71756400388881157594730855769, −4.99831323096596741061562333774, −4.35677212912785124425933507335, −3.20549955564044440374190588868, −1.88554741262939899031575816265, −0.57050348966162289203984343807, 1.98444053775260581052617990966, 2.73940557679093953040526027343, 4.11688513349309010716803009675, 4.55609198691533021635900829786, 5.52520182392698768644590709199, 6.61761197713007988636443911872, 7.32177067661748497866007543812, 8.202845236881996624845364879785, 8.847214791660957492482944447247, 9.339442571435666925162361956562

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