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Properties

Label 2-3525-141.53-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.587 + 0.808i$
Analytic cond. $1.75920$
Root an. cond. $1.32634$
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  + (−0.680 − 1.56i)2-s + (0.997 + 0.0682i)3-s + (−1.31 + 1.40i)4-s + (−0.572 − 1.61i)6-s + (1.48 + 0.526i)8-s + (0.990 + 0.136i)9-s + (−1.40 + 1.31i)12-s + (−0.0523 − 0.764i)16-s + (0.256 − 0.315i)17-s + (−0.461 − 1.64i)18-s + (−1.16 − 0.709i)19-s + (0.730 − 1.68i)23-s + (1.44 + 0.626i)24-s + (0.979 + 0.203i)27-s + (0.105 + 1.54i)31-s + (0.232 − 0.120i)32-s + ⋯
L(s)  = 1  + (−0.680 − 1.56i)2-s + (0.997 + 0.0682i)3-s + (−1.31 + 1.40i)4-s + (−0.572 − 1.61i)6-s + (1.48 + 0.526i)8-s + (0.990 + 0.136i)9-s + (−1.40 + 1.31i)12-s + (−0.0523 − 0.764i)16-s + (0.256 − 0.315i)17-s + (−0.461 − 1.64i)18-s + (−1.16 − 0.709i)19-s + (0.730 − 1.68i)23-s + (1.44 + 0.626i)24-s + (0.979 + 0.203i)27-s + (0.105 + 1.54i)31-s + (0.232 − 0.120i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.587 + 0.808i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (476, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.587 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188059666\)
\(L(\frac12)\) \(\approx\) \(1.188059666\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.997 - 0.0682i)T \)
5 \( 1 \)
47 \( 1 + (-0.816 + 0.576i)T \)
good2 \( 1 + (0.680 + 1.56i)T + (-0.682 + 0.730i)T^{2} \)
7 \( 1 + (-0.917 + 0.398i)T^{2} \)
11 \( 1 + (-0.203 + 0.979i)T^{2} \)
13 \( 1 + (0.854 + 0.519i)T^{2} \)
17 \( 1 + (-0.256 + 0.315i)T + (-0.203 - 0.979i)T^{2} \)
19 \( 1 + (1.16 + 0.709i)T + (0.460 + 0.887i)T^{2} \)
23 \( 1 + (-0.730 + 1.68i)T + (-0.682 - 0.730i)T^{2} \)
29 \( 1 + (-0.854 + 0.519i)T^{2} \)
31 \( 1 + (-0.105 - 1.54i)T + (-0.990 + 0.136i)T^{2} \)
37 \( 1 + (-0.334 - 0.942i)T^{2} \)
41 \( 1 + (0.775 - 0.631i)T^{2} \)
43 \( 1 + (-0.0682 - 0.997i)T^{2} \)
53 \( 1 + (-1.86 + 0.663i)T + (0.775 - 0.631i)T^{2} \)
59 \( 1 + (0.0682 - 0.997i)T^{2} \)
61 \( 1 + (1.11 + 1.57i)T + (-0.334 + 0.942i)T^{2} \)
67 \( 1 + (-0.917 - 0.398i)T^{2} \)
71 \( 1 + (-0.682 - 0.730i)T^{2} \)
73 \( 1 + (0.962 - 0.269i)T^{2} \)
79 \( 1 + (-0.713 - 1.37i)T + (-0.576 + 0.816i)T^{2} \)
83 \( 1 + (0.422 + 0.519i)T + (-0.203 + 0.979i)T^{2} \)
89 \( 1 + (-0.460 + 0.887i)T^{2} \)
97 \( 1 + (-0.990 - 0.136i)T^{2} \)
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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

−8.653824137973756783089552602434, −8.341649043335911873271981524957, −7.23346629360114794636390730859, −6.55988877315323736278990595323, −5.00186000929511296331290084412, −4.26940675674209184412888565139, −3.47489139252508768323997072361, −2.64372925425477687359866297965, −2.12193714782936190874351556762, −0.896073983635533367767483942710, 1.27366774440633551414931404206, 2.52315136304109892275950322575, 3.76885849303129369148983252379, 4.48802298064921835435632149860, 5.64592798278381218439851675263, 6.11508254557455895126865755577, 7.15378234764696357197904190641, 7.53475772687070677767959111700, 8.150188067864992024351916331436, 8.893734710835369439470694029176

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