Properties

Degree $2$
Conductor $690$
Sign $1$
Motivic weight $1$
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 + 4·7-s − 8-s + 9-s − 10-s + 2·11-s − 12-s + 4·13-s − 4·14-s − 15-s + 16-s − 6·17-s − 18-s − 4·19-s + 20-s − 4·21-s − 2·22-s − 23-s + 24-s + 25-s − 4·26-s − 27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s + 1.10·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{690} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.209835000\)
\(L(\frac12)\) \(\approx\) \(1.209835000\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−19.47539972490567, −18.73282160808822, −17.78033650384426, −17.58925452120263, −17.11837317912692, −15.86414497059792, −15.53638215021808, −14.35553000075517, −13.84282912902817, −12.71628240588627, −11.81248231312347, −11.00698200947200, −10.76553953718344, −9.611473329356770, −8.523347824630566, −8.248486289861665, −6.761216744730868, −6.287471888664362, −5.029258556565941, −4.142876720638264, −2.226633881216675, −1.181234601526733, 1.181234601526733, 2.226633881216675, 4.142876720638264, 5.029258556565941, 6.287471888664362, 6.761216744730868, 8.248486289861665, 8.523347824630566, 9.611473329356770, 10.76553953718344, 11.00698200947200, 11.81248231312347, 12.71628240588627, 13.84282912902817, 14.35553000075517, 15.53638215021808, 15.86414497059792, 17.11837317912692, 17.58925452120263, 17.78033650384426, 18.73282160808822, 19.47539972490567

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