Properties

Label 2-1710-1.1-c1-0-8
Degree $2$
Conductor $1710$
Sign $1$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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 + 4-s − 5-s − 2·7-s + 8-s − 10-s + 2·11-s − 2·14-s + 16-s + 2·17-s + 19-s − 20-s + 2·22-s + 8·23-s + 25-s − 2·28-s + 32-s + 2·34-s + 2·35-s + 4·37-s + 38-s − 40-s + 8·41-s − 6·43-s + 2·44-s + 8·46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.229·19-s − 0.223·20-s + 0.426·22-s + 1.66·23-s + 1/5·25-s − 0.377·28-s + 0.176·32-s + 0.342·34-s + 0.338·35-s + 0.657·37-s + 0.162·38-s − 0.158·40-s + 1.24·41-s − 0.914·43-s + 0.301·44-s + 1.17·46-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.469443372\)
\(L(\frac12)\) \(\approx\) \(2.469443372\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
19 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−9.337512026667636764455235469413, −8.576169949214302797369399942067, −7.47231387261999183818040048050, −6.92060556397032542472367252370, −6.07657152725821217624879228381, −5.22611296970313731007175215014, −4.24244004877095440204684144895, −3.45191506153810335863912261323, −2.63124068493624222532667688726, −1.02054089713911743443106437818, 1.02054089713911743443106437818, 2.63124068493624222532667688726, 3.45191506153810335863912261323, 4.24244004877095440204684144895, 5.22611296970313731007175215014, 6.07657152725821217624879228381, 6.92060556397032542472367252370, 7.47231387261999183818040048050, 8.576169949214302797369399942067, 9.337512026667636764455235469413

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