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Properties

Label 2-3525-141.17-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.575 + 0.817i$
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  + (0.672 − 0.628i)2-s + (−0.136 − 0.990i)3-s + (−0.0104 + 0.153i)4-s + (−0.713 − 0.580i)6-s + (0.669 + 0.823i)8-s + (−0.962 + 0.269i)9-s + (0.153 − 0.0104i)12-s + (0.815 + 0.112i)16-s + (1.79 + 0.373i)17-s + (−0.478 + 0.786i)18-s + (0.0627 − 0.121i)19-s + (0.997 + 0.931i)23-s + (0.724 − 0.775i)24-s + (0.398 + 0.917i)27-s + (−0.403 − 0.0554i)31-s + (−0.248 + 0.175i)32-s + ⋯
L(s)  = 1  + (0.672 − 0.628i)2-s + (−0.136 − 0.990i)3-s + (−0.0104 + 0.153i)4-s + (−0.713 − 0.580i)6-s + (0.669 + 0.823i)8-s + (−0.962 + 0.269i)9-s + (0.153 − 0.0104i)12-s + (0.815 + 0.112i)16-s + (1.79 + 0.373i)17-s + (−0.478 + 0.786i)18-s + (0.0627 − 0.121i)19-s + (0.997 + 0.931i)23-s + (0.724 − 0.775i)24-s + (0.398 + 0.917i)27-s + (−0.403 − 0.0554i)31-s + (−0.248 + 0.175i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.575 + 0.817i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (3401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.575 + 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.842472411\)
\(L(\frac12)\) \(\approx\) \(1.842472411\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.136 + 0.990i)T \)
5 \( 1 \)
47 \( 1 + (0.942 - 0.334i)T \)
good2 \( 1 + (-0.672 + 0.628i)T + (0.0682 - 0.997i)T^{2} \)
7 \( 1 + (0.682 + 0.730i)T^{2} \)
11 \( 1 + (0.917 - 0.398i)T^{2} \)
13 \( 1 + (0.460 - 0.887i)T^{2} \)
17 \( 1 + (-1.79 - 0.373i)T + (0.917 + 0.398i)T^{2} \)
19 \( 1 + (-0.0627 + 0.121i)T + (-0.576 - 0.816i)T^{2} \)
23 \( 1 + (-0.997 - 0.931i)T + (0.0682 + 0.997i)T^{2} \)
29 \( 1 + (-0.460 - 0.887i)T^{2} \)
31 \( 1 + (0.403 + 0.0554i)T + (0.962 + 0.269i)T^{2} \)
37 \( 1 + (-0.775 - 0.631i)T^{2} \)
41 \( 1 + (-0.203 - 0.979i)T^{2} \)
43 \( 1 + (-0.990 - 0.136i)T^{2} \)
53 \( 1 + (-1.21 + 1.49i)T + (-0.203 - 0.979i)T^{2} \)
59 \( 1 + (0.990 - 0.136i)T^{2} \)
61 \( 1 + (0.572 + 1.61i)T + (-0.775 + 0.631i)T^{2} \)
67 \( 1 + (0.682 - 0.730i)T^{2} \)
71 \( 1 + (0.0682 + 0.997i)T^{2} \)
73 \( 1 + (0.854 + 0.519i)T^{2} \)
79 \( 1 + (-0.234 - 0.332i)T + (-0.334 + 0.942i)T^{2} \)
83 \( 1 + (-1.51 + 0.315i)T + (0.917 - 0.398i)T^{2} \)
89 \( 1 + (0.576 - 0.816i)T^{2} \)
97 \( 1 + (0.962 - 0.269i)T^{2} \)
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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

−8.382003924918613980075197001574, −7.84202533665058992024964840819, −7.26461124585399248686858735477, −6.37526735173793622920628792715, −5.39095091612631282781915682787, −5.02112400957245437626807976967, −3.60362149241222674560984214677, −3.21513769981778055647309785146, −2.12923389985054139061170242797, −1.25309820597664109475659888295, 1.10559108428102560551285745541, 2.82365215228740030742931064342, 3.62549341362819155190913007258, 4.46184447450965334784009392795, 5.13906045462331687423235793728, 5.67649033545426055728236031763, 6.38741386047547230117125619517, 7.26938020716157728807630426840, 8.005615954555167296902248552841, 9.005881008086559122639651759829

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