Properties

Label 2-3570-1.1-c1-0-27
Degree $2$
Conductor $3570$
Sign $1$
Analytic cond. $28.5065$
Root an. cond. $5.33915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 4·11-s − 12-s + 6·13-s − 14-s − 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s + 21-s + 4·22-s − 24-s + 25-s + 6·26-s − 27-s − 28-s − 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3570\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(28.5065\)
Root analytic conductor: \(5.33915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3570} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3570,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.153643889\)
\(L(\frac12)\) \(\approx\) \(3.153643889\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.734118958187299957225954123695, −7.52160895362484104782086974891, −6.85726175005492809831884360687, −6.09670270857489217077302992275, −5.73803976573405992307271144607, −4.84330703048968464621470993606, −3.71577347919909976402126582234, −3.45054591198420993514937638569, −1.90234199540069152675428618399, −1.05541827782359798724099818115, 1.05541827782359798724099818115, 1.90234199540069152675428618399, 3.45054591198420993514937638569, 3.71577347919909976402126582234, 4.84330703048968464621470993606, 5.73803976573405992307271144607, 6.09670270857489217077302992275, 6.85726175005492809831884360687, 7.52160895362484104782086974891, 8.734118958187299957225954123695

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