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Properties

Label 2-3525-705.422-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.375 + 0.926i$
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  + (−0.147 + 0.147i)2-s + (−0.0523 + 0.998i)3-s + 0.956i·4-s + (−0.139 − 0.155i)6-s + (−0.575 − 0.575i)7-s + (−0.289 − 0.289i)8-s + (−0.994 − 0.104i)9-s + (−0.954 − 0.0500i)12-s + 0.170·14-s − 0.870·16-s + (−1.38 + 1.38i)17-s + (0.162 − 0.131i)18-s + (0.604 − 0.544i)21-s + (0.303 − 0.273i)24-s + (0.156 − 0.987i)27-s + (0.550 − 0.550i)28-s + ⋯
L(s)  = 1  + (−0.147 + 0.147i)2-s + (−0.0523 + 0.998i)3-s + 0.956i·4-s + (−0.139 − 0.155i)6-s + (−0.575 − 0.575i)7-s + (−0.289 − 0.289i)8-s + (−0.994 − 0.104i)9-s + (−0.954 − 0.0500i)12-s + 0.170·14-s − 0.870·16-s + (−1.38 + 1.38i)17-s + (0.162 − 0.131i)18-s + (0.604 − 0.544i)21-s + (0.303 − 0.273i)24-s + (0.156 − 0.987i)27-s + (0.550 − 0.550i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.375 + 0.926i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.375 + 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1987606266\)
\(L(\frac12)\) \(\approx\) \(0.1987606266\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0523 - 0.998i)T \)
5 \( 1 \)
47 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (0.147 - 0.147i)T - iT^{2} \)
7 \( 1 + (0.575 + 0.575i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.34 + 1.34i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
53 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
59 \( 1 + 0.813T + T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.48iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 0.618iT - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 + 1.17T + T^{2} \)
97 \( 1 + (-0.831 - 0.831i)T + iT^{2} \)
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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

−9.167058567890437119205023941432, −8.558687941570352077067812101854, −8.003127030011108165075181566618, −6.92052467058421177715624038369, −6.46975744401697797529890208248, −5.45360914895986763787602373472, −4.40000084247516593683179289781, −3.88237865758571050168718332295, −3.24918598732136589724199183327, −2.15857339121899320203573443297, 0.11496456367563553406290749908, 1.50121395611571877828492597121, 2.40849546425320038477681198445, 3.11180490356561801669165934372, 4.68139210362378468570954155002, 5.30804019542595828390424448605, 6.19811984790937303553207389775, 6.64801314127390673037512390732, 7.30840119317662539558692719680, 8.398335522686211830610943705181

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