Properties

Label 2-1710-1.1-c1-0-27
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 6·11-s − 4·14-s + 16-s + 4·17-s − 19-s + 20-s − 6·22-s − 4·23-s + 25-s − 4·28-s − 10·29-s − 2·31-s + 32-s + 4·34-s − 4·35-s − 4·37-s − 38-s + 40-s + 10·41-s − 12·43-s − 6·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s − 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.223·20-s − 1.27·22-s − 0.834·23-s + 1/5·25-s − 0.755·28-s − 1.85·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s − 0.676·35-s − 0.657·37-s − 0.162·38-s + 0.158·40-s + 1.56·41-s − 1.82·43-s − 0.904·44-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: \(1\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
19 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.100008625808348323448155408874, −7.88364453344063472224727462736, −7.32543187531103290631206584169, −6.24210546524596288177148842028, −5.73021129110967032737999380822, −4.97604743562511676044252184734, −3.66300986168232350310060029085, −3.01793818203876802699399218549, −2.03762284961702489869396909615, 0, 2.03762284961702489869396909615, 3.01793818203876802699399218549, 3.66300986168232350310060029085, 4.97604743562511676044252184734, 5.73021129110967032737999380822, 6.24210546524596288177148842028, 7.32543187531103290631206584169, 7.88364453344063472224727462736, 9.100008625808348323448155408874

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