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Properties

Label 2-3525-141.65-c0-0-1
Degree $2$
Conductor $3525$
Sign $0.968 + 0.248i$
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  + (1.11 + 0.787i)2-s + (0.631 − 0.775i)3-s + (0.289 + 0.813i)4-s + (1.31 − 0.368i)6-s + (0.0502 − 0.179i)8-s + (−0.203 − 0.979i)9-s + (0.813 + 0.289i)12-s + (0.867 − 0.705i)16-s + (0.478 − 0.786i)17-s + (0.543 − 1.25i)18-s + (0.457 − 0.489i)19-s + (−0.942 + 0.665i)23-s + (−0.107 − 0.152i)24-s + (−0.887 − 0.460i)27-s + (−1.32 + 1.07i)31-s + (1.33 − 0.0914i)32-s + ⋯
L(s)  = 1  + (1.11 + 0.787i)2-s + (0.631 − 0.775i)3-s + (0.289 + 0.813i)4-s + (1.31 − 0.368i)6-s + (0.0502 − 0.179i)8-s + (−0.203 − 0.979i)9-s + (0.813 + 0.289i)12-s + (0.867 − 0.705i)16-s + (0.478 − 0.786i)17-s + (0.543 − 1.25i)18-s + (0.457 − 0.489i)19-s + (−0.942 + 0.665i)23-s + (−0.107 − 0.152i)24-s + (−0.887 − 0.460i)27-s + (−1.32 + 1.07i)31-s + (1.33 − 0.0914i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (3026, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.834523218\)
\(L(\frac12)\) \(\approx\) \(2.834523218\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.631 + 0.775i)T \)
5 \( 1 \)
47 \( 1 + (0.136 - 0.990i)T \)
good2 \( 1 + (-1.11 - 0.787i)T + (0.334 + 0.942i)T^{2} \)
7 \( 1 + (-0.576 + 0.816i)T^{2} \)
11 \( 1 + (-0.460 + 0.887i)T^{2} \)
13 \( 1 + (0.682 - 0.730i)T^{2} \)
17 \( 1 + (-0.478 + 0.786i)T + (-0.460 - 0.887i)T^{2} \)
19 \( 1 + (-0.457 + 0.489i)T + (-0.0682 - 0.997i)T^{2} \)
23 \( 1 + (0.942 - 0.665i)T + (0.334 - 0.942i)T^{2} \)
29 \( 1 + (-0.682 - 0.730i)T^{2} \)
31 \( 1 + (1.32 - 1.07i)T + (0.203 - 0.979i)T^{2} \)
37 \( 1 + (0.962 - 0.269i)T^{2} \)
41 \( 1 + (-0.854 + 0.519i)T^{2} \)
43 \( 1 + (-0.775 + 0.631i)T^{2} \)
53 \( 1 + (-0.109 - 0.391i)T + (-0.854 + 0.519i)T^{2} \)
59 \( 1 + (0.775 + 0.631i)T^{2} \)
61 \( 1 + (-1.81 - 0.249i)T + (0.962 + 0.269i)T^{2} \)
67 \( 1 + (-0.576 - 0.816i)T^{2} \)
71 \( 1 + (0.334 - 0.942i)T^{2} \)
73 \( 1 + (-0.917 - 0.398i)T^{2} \)
79 \( 1 + (-0.116 - 1.70i)T + (-0.990 + 0.136i)T^{2} \)
83 \( 1 + (-1.00 - 1.64i)T + (-0.460 + 0.887i)T^{2} \)
89 \( 1 + (0.0682 - 0.997i)T^{2} \)
97 \( 1 + (0.203 + 0.979i)T^{2} \)
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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

−8.435603323552143336897418947980, −7.67168846848375546997459032255, −7.12234589151051699517699187677, −6.59761495818628830919762800758, −5.62985213647010558648702844665, −5.18953761595840907002993822137, −4.01622821860471635508877664972, −3.41185306401591665342067301361, −2.48169326821975565441226378134, −1.17573243363722769905545644442, 1.80330535969915277955714449787, 2.50297429910564706357248780625, 3.61030482038160136174684158101, 3.83414403017179164269399586662, 4.74529656114204123635064441461, 5.50546075608983020551596999477, 6.12378529739656113517603042734, 7.49993178515985438125869414777, 8.107743830317001427934732646034, 8.810115055045751935230157880051

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