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Properties

Label 2-3525-705.167-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.841 - 0.540i$
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  + (−1.74 − 0.554i)2-s + (−0.903 + 0.429i)3-s + (1.93 + 1.36i)4-s + (1.81 − 0.249i)6-s + (−1.51 − 1.99i)8-s + (0.631 − 0.775i)9-s + (−2.33 − 0.401i)12-s + (0.746 + 2.09i)16-s + (1.48 − 0.837i)17-s + (−1.53 + 1.00i)18-s + (−1.49 + 0.650i)19-s + (0.537 + 1.69i)23-s + (2.22 + 1.15i)24-s + (−0.236 + 0.971i)27-s + (−0.508 + 0.180i)31-s + (−0.0540 − 1.58i)32-s + ⋯
L(s)  = 1  + (−1.74 − 0.554i)2-s + (−0.903 + 0.429i)3-s + (1.93 + 1.36i)4-s + (1.81 − 0.249i)6-s + (−1.51 − 1.99i)8-s + (0.631 − 0.775i)9-s + (−2.33 − 0.401i)12-s + (0.746 + 2.09i)16-s + (1.48 − 0.837i)17-s + (−1.53 + 1.00i)18-s + (−1.49 + 0.650i)19-s + (0.537 + 1.69i)23-s + (2.22 + 1.15i)24-s + (−0.236 + 0.971i)27-s + (−0.508 + 0.180i)31-s + (−0.0540 − 1.58i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.841 - 0.540i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (2282, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.841 - 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3632071122\)
\(L(\frac12)\) \(\approx\) \(0.3632071122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.903 - 0.429i)T \)
5 \( 1 \)
47 \( 1 + (0.657 + 0.753i)T \)
good2 \( 1 + (1.74 + 0.554i)T + (0.816 + 0.576i)T^{2} \)
7 \( 1 + (-0.887 + 0.460i)T^{2} \)
11 \( 1 + (0.854 - 0.519i)T^{2} \)
13 \( 1 + (-0.398 - 0.917i)T^{2} \)
17 \( 1 + (-1.48 + 0.837i)T + (0.519 - 0.854i)T^{2} \)
19 \( 1 + (1.49 - 0.650i)T + (0.682 - 0.730i)T^{2} \)
23 \( 1 + (-0.537 - 1.69i)T + (-0.816 + 0.576i)T^{2} \)
29 \( 1 + (0.917 + 0.398i)T^{2} \)
31 \( 1 + (0.508 - 0.180i)T + (0.775 - 0.631i)T^{2} \)
37 \( 1 + (-0.136 - 0.990i)T^{2} \)
41 \( 1 + (0.962 - 0.269i)T^{2} \)
43 \( 1 + (0.942 - 0.334i)T^{2} \)
53 \( 1 + (-1.23 - 0.937i)T + (0.269 + 0.962i)T^{2} \)
59 \( 1 + (-0.334 + 0.942i)T^{2} \)
61 \( 1 + (-0.0277 + 0.405i)T + (-0.990 - 0.136i)T^{2} \)
67 \( 1 + (-0.887 - 0.460i)T^{2} \)
71 \( 1 + (0.576 + 0.816i)T^{2} \)
73 \( 1 + (0.979 + 0.203i)T^{2} \)
79 \( 1 + (-1.40 - 1.31i)T + (0.0682 + 0.997i)T^{2} \)
83 \( 1 + (1.72 + 0.971i)T + (0.519 + 0.854i)T^{2} \)
89 \( 1 + (0.682 + 0.730i)T^{2} \)
97 \( 1 + (0.631 - 0.775i)T^{2} \)
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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

−9.038901801815144980887563515408, −8.241713644540313151384959767200, −7.41098673130509769339002110100, −6.94857003376062799698428707581, −5.93454387392936183114374465952, −5.22455020967162302986894662804, −3.91058961615291926925832210982, −3.16115664134441489168936848264, −1.88914348589658687854520302793, −0.920306813370384182342501602410, 0.58067553246400017821339540392, 1.63566417755865334043324244868, 2.57607882345684194639631383620, 4.23475693708201684584315585099, 5.33279720530848804027094616075, 6.08136674099743702344999917508, 6.62901575466018023304156298363, 7.23365769946211541266443521848, 8.051943877186038710076650488740, 8.496845618065913239747718722184

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