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Properties

Label 2-3525-705.218-c0-0-0
Degree $2$
Conductor $3525$
Sign $-0.869 + 0.493i$
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  + (−0.0314 + 0.919i)2-s + (−0.604 + 0.796i)3-s + (0.153 + 0.0104i)4-s + (−0.713 − 0.580i)6-s + (−0.108 + 1.05i)8-s + (−0.269 − 0.962i)9-s + (−0.100 + 0.115i)12-s + (−0.815 − 0.112i)16-s + (−1.53 + 1.00i)17-s + (0.893 − 0.217i)18-s + (−0.917 + 1.77i)19-s + (1.46 − 0.0499i)23-s + (−0.775 − 0.724i)24-s + (0.930 + 0.366i)27-s + (0.266 − 1.93i)31-s + (−0.0516 + 0.299i)32-s + ⋯
L(s)  = 1  + (−0.0314 + 0.919i)2-s + (−0.604 + 0.796i)3-s + (0.153 + 0.0104i)4-s + (−0.713 − 0.580i)6-s + (−0.108 + 1.05i)8-s + (−0.269 − 0.962i)9-s + (−0.100 + 0.115i)12-s + (−0.815 − 0.112i)16-s + (−1.53 + 1.00i)17-s + (0.893 − 0.217i)18-s + (−0.917 + 1.77i)19-s + (1.46 − 0.0499i)23-s + (−0.775 − 0.724i)24-s + (0.930 + 0.366i)27-s + (0.266 − 1.93i)31-s + (−0.0516 + 0.299i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.869 + 0.493i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.869 + 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7965804825\)
\(L(\frac12)\) \(\approx\) \(0.7965804825\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.604 - 0.796i)T \)
5 \( 1 \)
47 \( 1 + (0.429 - 0.903i)T \)
good2 \( 1 + (0.0314 - 0.919i)T + (-0.997 - 0.0682i)T^{2} \)
7 \( 1 + (0.730 - 0.682i)T^{2} \)
11 \( 1 + (-0.917 + 0.398i)T^{2} \)
13 \( 1 + (-0.887 - 0.460i)T^{2} \)
17 \( 1 + (1.53 - 1.00i)T + (0.398 - 0.917i)T^{2} \)
19 \( 1 + (0.917 - 1.77i)T + (-0.576 - 0.816i)T^{2} \)
23 \( 1 + (-1.46 + 0.0499i)T + (0.997 - 0.0682i)T^{2} \)
29 \( 1 + (-0.460 - 0.887i)T^{2} \)
31 \( 1 + (-0.266 + 1.93i)T + (-0.962 - 0.269i)T^{2} \)
37 \( 1 + (-0.631 + 0.775i)T^{2} \)
41 \( 1 + (0.203 + 0.979i)T^{2} \)
43 \( 1 + (-0.136 + 0.990i)T^{2} \)
53 \( 1 + (1.91 - 0.196i)T + (0.979 - 0.203i)T^{2} \)
59 \( 1 + (-0.990 + 0.136i)T^{2} \)
61 \( 1 + (-0.572 - 1.61i)T + (-0.775 + 0.631i)T^{2} \)
67 \( 1 + (0.730 + 0.682i)T^{2} \)
71 \( 1 + (0.0682 + 0.997i)T^{2} \)
73 \( 1 + (0.519 - 0.854i)T^{2} \)
79 \( 1 + (0.332 - 0.234i)T + (0.334 - 0.942i)T^{2} \)
83 \( 1 + (1.29 + 0.850i)T + (0.398 + 0.917i)T^{2} \)
89 \( 1 + (-0.576 + 0.816i)T^{2} \)
97 \( 1 + (-0.269 - 0.962i)T^{2} \)
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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

−9.005965377850193918312341035426, −8.400438722849086642638790479661, −7.68998584498577904476385603150, −6.69400603505240574048372549794, −6.15559505430485274575801052668, −5.73431174176967057813855878528, −4.63491443990454371346165687092, −4.13893347191703254320913324914, −2.95586958638775910045308396741, −1.77346018852563400818396528008, 0.47415932031245138295633621478, 1.68912541032057603756836089061, 2.52783905947530774772684434013, 3.20097167275082321451214061368, 4.70055882835689345455664063131, 4.99509761666968436194073536349, 6.46317262638246120336154634361, 6.77770612961223436942243700843, 7.23693680314669176379923407261, 8.520284540470168348231024701403

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