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Properties

Label 2-3525-141.98-c0-0-0
Degree $2$
Conductor $3525$
Sign $-0.959 + 0.280i$
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  + (−0.418 + 1.49i)2-s + (0.887 + 0.460i)3-s + (−1.20 − 0.731i)4-s + (−1.05 + 1.13i)6-s + (0.461 − 0.430i)8-s + (0.576 + 0.816i)9-s + (−0.731 − 1.20i)12-s + (−0.196 − 0.379i)16-s + (−1.97 − 0.135i)17-s + (−1.46 + 0.519i)18-s + (1.32 + 1.07i)19-s + (0.519 + 1.85i)23-s + (0.607 − 0.170i)24-s + (0.136 + 0.990i)27-s + (−0.0627 − 0.121i)31-s + (1.26 − 0.263i)32-s + ⋯
L(s)  = 1  + (−0.418 + 1.49i)2-s + (0.887 + 0.460i)3-s + (−1.20 − 0.731i)4-s + (−1.05 + 1.13i)6-s + (0.461 − 0.430i)8-s + (0.576 + 0.816i)9-s + (−0.731 − 1.20i)12-s + (−0.196 − 0.379i)16-s + (−1.97 − 0.135i)17-s + (−1.46 + 0.519i)18-s + (1.32 + 1.07i)19-s + (0.519 + 1.85i)23-s + (0.607 − 0.170i)24-s + (0.136 + 0.990i)27-s + (−0.0627 − 0.121i)31-s + (1.26 − 0.263i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.959 + 0.280i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.959 + 0.280i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159674753\)
\(L(\frac12)\) \(\approx\) \(1.159674753\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.887 - 0.460i)T \)
5 \( 1 \)
47 \( 1 + (0.398 - 0.917i)T \)
good2 \( 1 + (0.418 - 1.49i)T + (-0.854 - 0.519i)T^{2} \)
7 \( 1 + (0.962 + 0.269i)T^{2} \)
11 \( 1 + (0.990 - 0.136i)T^{2} \)
13 \( 1 + (-0.775 - 0.631i)T^{2} \)
17 \( 1 + (1.97 + 0.135i)T + (0.990 + 0.136i)T^{2} \)
19 \( 1 + (-1.32 - 1.07i)T + (0.203 + 0.979i)T^{2} \)
23 \( 1 + (-0.519 - 1.85i)T + (-0.854 + 0.519i)T^{2} \)
29 \( 1 + (0.775 - 0.631i)T^{2} \)
31 \( 1 + (0.0627 + 0.121i)T + (-0.576 + 0.816i)T^{2} \)
37 \( 1 + (0.682 - 0.730i)T^{2} \)
41 \( 1 + (0.0682 + 0.997i)T^{2} \)
43 \( 1 + (0.460 + 0.887i)T^{2} \)
53 \( 1 + (0.842 + 0.787i)T + (0.0682 + 0.997i)T^{2} \)
59 \( 1 + (-0.460 + 0.887i)T^{2} \)
61 \( 1 + (-0.614 - 0.266i)T + (0.682 + 0.730i)T^{2} \)
67 \( 1 + (0.962 - 0.269i)T^{2} \)
71 \( 1 + (-0.854 + 0.519i)T^{2} \)
73 \( 1 + (-0.334 - 0.942i)T^{2} \)
79 \( 1 + (-0.0277 - 0.133i)T + (-0.917 + 0.398i)T^{2} \)
83 \( 1 + (-1.36 + 0.0931i)T + (0.990 - 0.136i)T^{2} \)
89 \( 1 + (-0.203 + 0.979i)T^{2} \)
97 \( 1 + (-0.576 - 0.816i)T^{2} \)
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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

−9.175064286964353718854072900276, −8.215409473358342318517940239513, −7.76429764404623060072363764057, −7.10679991455040388095032078469, −6.38799461213744941092439383492, −5.40775792993050732463122357773, −4.85283890256157266079579133455, −3.85760948051125304814504623744, −2.94684443577806208312720666310, −1.69914600827643637255989993444, 0.69296582745663406915560762157, 1.88186126530679500133978831371, 2.59522531296592496098002134352, 3.19507984476501833902733079859, 4.20839275657214682103739904587, 4.85883879555320821678978371016, 6.48170843339893894573324563044, 6.85418483504286200689748753271, 7.935432079798046138378149470833, 8.723635807359180417338067764467

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