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Properties

Label 2-3525-705.563-c0-0-4
Degree $2$
Conductor $3525$
Sign $0.925 + 0.379i$
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  + (−0.437 − 0.437i)2-s + (0.156 − 0.987i)3-s − 0.618i·4-s + (−0.5 + 0.363i)6-s + (−1.34 + 1.34i)7-s + (−0.707 + 0.707i)8-s + (−0.951 − 0.309i)9-s + (−0.610 − 0.0966i)12-s + 1.17·14-s + (1.14 + 1.14i)17-s + (0.280 + 0.550i)18-s + (1.11 + 1.53i)21-s + (0.587 + 0.809i)24-s + (−0.453 + 0.891i)27-s + (0.831 + 0.831i)28-s + ⋯
L(s)  = 1  + (−0.437 − 0.437i)2-s + (0.156 − 0.987i)3-s − 0.618i·4-s + (−0.5 + 0.363i)6-s + (−1.34 + 1.34i)7-s + (−0.707 + 0.707i)8-s + (−0.951 − 0.309i)9-s + (−0.610 − 0.0966i)12-s + 1.17·14-s + (1.14 + 1.14i)17-s + (0.280 + 0.550i)18-s + (1.11 + 1.53i)21-s + (0.587 + 0.809i)24-s + (−0.453 + 0.891i)27-s + (0.831 + 0.831i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.925 + 0.379i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.925 + 0.379i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6964811176\)
\(L(\frac12)\) \(\approx\) \(0.6964811176\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.156 + 0.987i)T \)
5 \( 1 \)
47 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.437 + 0.437i)T + iT^{2} \)
7 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-1.14 - 1.14i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.831 + 0.831i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (0.437 - 0.437i)T - iT^{2} \)
59 \( 1 - 1.90T + T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.17iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.61iT - T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
89 \( 1 - 1.90T + T^{2} \)
97 \( 1 + (-1.34 + 1.34i)T - iT^{2} \)
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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

−8.724053593114718275978740145597, −8.225972256960089051556904946071, −7.19065069157174873049783695556, −6.27664185233413630695292968716, −5.89571406624626333282911278228, −5.36071828801022992488450175599, −3.69372491733961460677505595423, −2.76832777737664046495113698413, −2.19931726441134502306091423299, −1.05980183717355881867326324300, 0.54417000979890595342875622833, 2.82224850602186634712397304182, 3.37236765352892535351321806077, 3.98588589379794116541402908576, 4.86665899404131015732134258080, 5.97341055158177950804155876579, 6.70599583743874167321276727945, 7.45414273645507499160524443051, 7.940056051154505960759162032157, 8.973451687893330493347251594424

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