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Properties

Label 2-3525-705.182-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.891 - 0.452i$
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  + (0.626 − 1.58i)2-s + (0.753 − 0.657i)3-s + (−1.40 − 1.31i)4-s + (−0.572 − 1.61i)6-s + (−1.41 + 0.675i)8-s + (0.136 − 0.990i)9-s + (−1.91 − 0.0655i)12-s + (0.0523 + 0.764i)16-s + (0.404 − 0.0416i)17-s + (−1.48 − 0.837i)18-s + (−1.24 − 0.759i)19-s + (−0.741 + 0.292i)23-s + (−0.626 + 1.44i)24-s + (−0.548 − 0.836i)27-s + (1.25 − 0.0861i)31-s + (−0.249 − 0.0792i)32-s + ⋯
L(s)  = 1  + (0.626 − 1.58i)2-s + (0.753 − 0.657i)3-s + (−1.40 − 1.31i)4-s + (−0.572 − 1.61i)6-s + (−1.41 + 0.675i)8-s + (0.136 − 0.990i)9-s + (−1.91 − 0.0655i)12-s + (0.0523 + 0.764i)16-s + (0.404 − 0.0416i)17-s + (−1.48 − 0.837i)18-s + (−1.24 − 0.759i)19-s + (−0.741 + 0.292i)23-s + (−0.626 + 1.44i)24-s + (−0.548 − 0.836i)27-s + (1.25 − 0.0861i)31-s + (−0.249 − 0.0792i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.891 - 0.452i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.891 - 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.984918216\)
\(L(\frac12)\) \(\approx\) \(1.984918216\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.753 + 0.657i)T \)
5 \( 1 \)
47 \( 1 + (0.985 + 0.169i)T \)
good2 \( 1 + (-0.626 + 1.58i)T + (-0.730 - 0.682i)T^{2} \)
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 \( 1 + (-0.404 + 0.0416i)T + (0.979 - 0.203i)T^{2} \)
19 \( 1 + (1.24 + 0.759i)T + (0.460 + 0.887i)T^{2} \)
23 \( 1 + (0.741 - 0.292i)T + (0.730 - 0.682i)T^{2} \)
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 \( 1 + (-0.850 + 1.78i)T + (-0.631 - 0.775i)T^{2} \)
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 \( 1 + (0.666 + 0.0684i)T + (0.979 + 0.203i)T^{2} \)
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

−8.545195935515454886436110810161, −7.74338415590188251487855628311, −6.77911767102172255819222917014, −6.00179515098952242450890927875, −4.93715131469690867789257541310, −4.13050549172665823663062128340, −3.44553382242590595975414830824, −2.55762552774367004901242419633, −1.99445137811502001781798351846, −0.866226733354597170799166233405, 2.07067801081609542517248025855, 3.27500176649205662784511222644, 4.09359655676674339125997458143, 4.59207869188211904395059319799, 5.47101545709148170215920839370, 6.17224099661313316430148791890, 6.90910559138148681816054987339, 7.78283453550495822255979573643, 8.308294316367705443639153407204, 8.706491048449002089559525842679

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