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Properties

Label 2-3525-705.227-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.803 + 0.594i$
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  + (−1.58 + 0.626i)2-s + (0.657 − 0.753i)3-s + (1.40 − 1.31i)4-s + (−0.572 + 1.61i)6-s + (−0.675 + 1.41i)8-s + (−0.136 − 0.990i)9-s + (−0.0655 − 1.91i)12-s + (0.0523 − 0.764i)16-s + (−0.0416 + 0.404i)17-s + (0.837 + 1.48i)18-s + (1.24 − 0.759i)19-s + (−0.292 + 0.741i)23-s + (0.626 + 1.44i)24-s + (−0.836 − 0.548i)27-s + (1.25 + 0.0861i)31-s + (−0.0792 − 0.249i)32-s + ⋯
L(s)  = 1  + (−1.58 + 0.626i)2-s + (0.657 − 0.753i)3-s + (1.40 − 1.31i)4-s + (−0.572 + 1.61i)6-s + (−0.675 + 1.41i)8-s + (−0.136 − 0.990i)9-s + (−0.0655 − 1.91i)12-s + (0.0523 − 0.764i)16-s + (−0.0416 + 0.404i)17-s + (0.837 + 1.48i)18-s + (1.24 − 0.759i)19-s + (−0.292 + 0.741i)23-s + (0.626 + 1.44i)24-s + (−0.836 − 0.548i)27-s + (1.25 + 0.0861i)31-s + (−0.0792 − 0.249i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.803 + 0.594i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (932, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.803 + 0.594i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7466852670\)
\(L(\frac12)\) \(\approx\) \(0.7466852670\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.657 + 0.753i)T \)
5 \( 1 \)
47 \( 1 + (0.169 + 0.985i)T \)
good2 \( 1 + (1.58 - 0.626i)T + (0.730 - 0.682i)T^{2} \)
7 \( 1 + (0.398 - 0.917i)T^{2} \)
11 \( 1 + (0.203 + 0.979i)T^{2} \)
13 \( 1 + (0.519 + 0.854i)T^{2} \)
17 \( 1 + (0.0416 - 0.404i)T + (-0.979 - 0.203i)T^{2} \)
19 \( 1 + (-1.24 + 0.759i)T + (0.460 - 0.887i)T^{2} \)
23 \( 1 + (0.292 - 0.741i)T + (-0.730 - 0.682i)T^{2} \)
29 \( 1 + (-0.854 - 0.519i)T^{2} \)
31 \( 1 + (-1.25 - 0.0861i)T + (0.990 + 0.136i)T^{2} \)
37 \( 1 + (-0.942 - 0.334i)T^{2} \)
41 \( 1 + (-0.775 - 0.631i)T^{2} \)
43 \( 1 + (-0.997 - 0.0682i)T^{2} \)
53 \( 1 + (-1.78 + 0.850i)T + (0.631 - 0.775i)T^{2} \)
59 \( 1 + (-0.0682 - 0.997i)T^{2} \)
61 \( 1 + (-1.11 + 1.57i)T + (-0.334 - 0.942i)T^{2} \)
67 \( 1 + (0.398 + 0.917i)T^{2} \)
71 \( 1 + (-0.682 + 0.730i)T^{2} \)
73 \( 1 + (-0.269 + 0.962i)T^{2} \)
79 \( 1 + (1.37 + 0.713i)T + (0.576 + 0.816i)T^{2} \)
83 \( 1 + (-0.0684 - 0.666i)T + (-0.979 + 0.203i)T^{2} \)
89 \( 1 + (0.460 + 0.887i)T^{2} \)
97 \( 1 + (-0.136 - 0.990i)T^{2} \)
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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

−8.525721840608458059619451959431, −8.038145009281849715591654911908, −7.35344740150168197383411421971, −6.82819304838670941780458924504, −6.13479093743992954134886540535, −5.23331471202617603089871142840, −3.80500798645435396895394074348, −2.74745863689136132895909621490, −1.76249754901119669400032578678, −0.815836926842903918684697467577, 1.14842369649638856449191412808, 2.34336706764663175766157134758, 2.95968910904503987624081399578, 3.88811912854328227907500851464, 4.88020293917875884909176084621, 5.89931098310917837456713275337, 7.12530580574975275518010105030, 7.64852443931459423321090722772, 8.466197643212784246244658763137, 8.768425898412827687956290770529

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