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Properties

Label 2-3525-705.248-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.3300881249\)
\(L(\frac12)\) \(\approx\) \(0.3300881249\)
\(L(1)\) 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

−8.365678155774378400633022971387, −7.73148403248733029523790643117, −7.03774518093822977757552025298, −6.42685844243203897046263296053, −6.00849098375699035198672172659, −5.03483164290693310477927553955, −4.16461089748968500333243081552, −2.71009061886826476760636972290, −2.01341496154873115330739437980, −0.18464350462171944694154680381, 1.71715781643951698010229602804, 2.49378087487990674581321457388, 3.44703198064536281926452957333, 4.13717241362247912983926121817, 4.67380062743891943160819277927, 6.00354162137261301475741654794, 6.41885942429003650388861972712, 7.86988199768711573456801546775, 8.545749716253460432676641012827, 9.101158704585942788166079172586

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