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Properties

Label 2-3525-705.248-c0-0-0
Degree $2$
Conductor $3525$
Sign $-0.0644 + 0.997i$
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  + (0.231 − 1.34i)2-s + (−0.102 + 0.994i)3-s + (−0.813 − 0.289i)4-s + (1.31 + 0.368i)6-s + (0.0913 − 0.162i)8-s + (−0.979 − 0.203i)9-s + (0.370 − 0.779i)12-s + (−0.867 − 0.705i)16-s + (0.893 + 0.217i)17-s + (−0.500 + 1.26i)18-s + (−1.28 − 1.37i)19-s + (1.61 − 0.277i)23-s + (0.152 + 0.107i)24-s + (0.302 − 0.953i)27-s + (0.655 − 0.806i)31-s + (−1.00 + 0.880i)32-s + ⋯
L(s)  = 1  + (0.231 − 1.34i)2-s + (−0.102 + 0.994i)3-s + (−0.813 − 0.289i)4-s + (1.31 + 0.368i)6-s + (0.0913 − 0.162i)8-s + (−0.979 − 0.203i)9-s + (0.370 − 0.779i)12-s + (−0.867 − 0.705i)16-s + (0.893 + 0.217i)17-s + (−0.500 + 1.26i)18-s + (−1.28 − 1.37i)19-s + (1.61 − 0.277i)23-s + (0.152 + 0.107i)24-s + (0.302 − 0.953i)27-s + (0.655 − 0.806i)31-s + (−1.00 + 0.880i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0644 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0644 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.0644 + 0.997i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (3068, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.0644 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.356062619\)
\(L(\frac12)\) \(\approx\) \(1.356062619\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.102 - 0.994i)T \)
5 \( 1 \)
47 \( 1 + (-0.796 - 0.604i)T \)
good2 \( 1 + (-0.231 + 1.34i)T + (-0.942 - 0.334i)T^{2} \)
7 \( 1 + (-0.816 + 0.576i)T^{2} \)
11 \( 1 + (0.460 + 0.887i)T^{2} \)
13 \( 1 + (0.730 - 0.682i)T^{2} \)
17 \( 1 + (-0.893 - 0.217i)T + (0.887 + 0.460i)T^{2} \)
19 \( 1 + (1.28 + 1.37i)T + (-0.0682 + 0.997i)T^{2} \)
23 \( 1 + (-1.61 + 0.277i)T + (0.942 - 0.334i)T^{2} \)
29 \( 1 + (-0.682 + 0.730i)T^{2} \)
31 \( 1 + (-0.655 + 0.806i)T + (-0.203 - 0.979i)T^{2} \)
37 \( 1 + (0.269 - 0.962i)T^{2} \)
41 \( 1 + (0.854 + 0.519i)T^{2} \)
43 \( 1 + (-0.631 + 0.775i)T^{2} \)
53 \( 1 + (-0.354 + 0.199i)T + (0.519 - 0.854i)T^{2} \)
59 \( 1 + (-0.775 + 0.631i)T^{2} \)
61 \( 1 + (1.81 - 0.249i)T + (0.962 - 0.269i)T^{2} \)
67 \( 1 + (-0.816 - 0.576i)T^{2} \)
71 \( 1 + (0.334 + 0.942i)T^{2} \)
73 \( 1 + (0.398 + 0.917i)T^{2} \)
79 \( 1 + (-1.70 - 0.116i)T + (0.990 + 0.136i)T^{2} \)
83 \( 1 + (-1.87 + 0.455i)T + (0.887 - 0.460i)T^{2} \)
89 \( 1 + (-0.0682 - 0.997i)T^{2} \)
97 \( 1 + (-0.979 - 0.203i)T^{2} \)
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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

−9.112612119685151716711485205920, −8.060689696380389302637820877933, −7.00446492544310081801020024353, −6.16896531682242783503365939614, −5.09696480936256303918180980501, −4.54112363019015904890423061593, −3.80139941272698232660774497561, −2.95748035932602485957551264514, −2.33680161414126865518920769361, −0.822597063330629542295847517506, 1.31462548944117887787011537001, 2.43789123664163873537881047912, 3.57246610368616481335119001128, 4.75780315085429087490095709781, 5.46234608765193486243063866697, 6.13056314332631382020072334274, 6.68743466457398522753064929696, 7.41871312044326073027817693891, 7.917284867611262067046890080257, 8.586907510266980543809303284920

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