Properties

Degree $2$
Conductor $1710$
Sign $-1$
Motivic weight $1$
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 − 8-s − 10-s − 4·11-s + 2·13-s + 16-s − 2·17-s − 19-s + 20-s + 4·22-s − 4·23-s + 25-s − 2·26-s − 6·29-s + 4·31-s − 32-s + 2·34-s − 6·37-s + 38-s − 40-s − 10·41-s − 4·43-s − 4·44-s + 4·46-s + 12·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.229·19-s + 0.223·20-s + 0.852·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s − 0.986·37-s + 0.162·38-s − 0.158·40-s − 1.56·41-s − 0.609·43-s − 0.603·44-s + 0.589·46-s + 1.75·47-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{1710} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.06215194266271, −18.80256639499208, −17.98462936840671, −17.59829327861164, −16.82120887677637, −16.09849463086131, −15.55103526163671, −14.93171325964662, −13.89473026163578, −13.36330734214120, −12.65058154995271, −11.76636966414295, −11.00607247937870, −10.32211283652082, −9.835786381192900, −8.822006495534931, −8.302227173011843, −7.465547427177268, −6.610744317155073, −5.790683230305019, −4.983395869020084, −3.712332457307571, −2.586457635742922, −1.664706721770475, 0, 1.664706721770475, 2.586457635742922, 3.712332457307571, 4.983395869020084, 5.790683230305019, 6.610744317155073, 7.465547427177268, 8.302227173011843, 8.822006495534931, 9.835786381192900, 10.32211283652082, 11.00607247937870, 11.76636966414295, 12.65058154995271, 13.36330734214120, 13.89473026163578, 14.93171325964662, 15.55103526163671, 16.09849463086131, 16.82120887677637, 17.59829327861164, 17.98462936840671, 18.80256639499208, 19.06215194266271

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