Properties

Label 2-3525-1.1-c1-0-132
Degree $2$
Conductor $3525$
Sign $-1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s − 2·6-s + 3·7-s + 9-s − 5·11-s − 2·12-s − 2·13-s + 6·14-s − 4·16-s + 6·17-s + 2·18-s − 6·19-s − 3·21-s − 10·22-s − 9·23-s − 4·26-s − 27-s + 6·28-s + 29-s − 2·31-s − 8·32-s + 5·33-s + 12·34-s + 2·36-s − 37-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1.13·7-s + 1/3·9-s − 1.50·11-s − 0.577·12-s − 0.554·13-s + 1.60·14-s − 16-s + 1.45·17-s + 0.471·18-s − 1.37·19-s − 0.654·21-s − 2.13·22-s − 1.87·23-s − 0.784·26-s − 0.192·27-s + 1.13·28-s + 0.185·29-s − 0.359·31-s − 1.41·32-s + 0.870·33-s + 2.05·34-s + 1/3·36-s − 0.164·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-1$
Analytic conductor: \(28.1472\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
47 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−7.83234803660229109588865904558, −7.54740476343963674549472631818, −6.24357414292211393153596697451, −5.78180778565250228622005721493, −5.02594328209869903685800983902, −4.60956624547012436038730148171, −3.74543273200017646223279827807, −2.64397051882091468672200773297, −1.82835020575181270599988730084, 0, 1.82835020575181270599988730084, 2.64397051882091468672200773297, 3.74543273200017646223279827807, 4.60956624547012436038730148171, 5.02594328209869903685800983902, 5.78180778565250228622005721493, 6.24357414292211393153596697451, 7.54740476343963674549472631818, 7.83234803660229109588865904558

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