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 + 8-s + 9-s + 10-s + 4·11-s + 12-s − 2·13-s + 15-s + 16-s − 6·17-s + 18-s + 4·19-s + 20-s + 4·22-s − 23-s + 24-s + 25-s − 2·26-s + 27-s − 2·29-s + 30-s + 32-s + 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-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 + 0.852·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.371·29-s + 0.182·30-s + 0.176·32-s + 0.696·33-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

\(L(1)\) \(\approx\) \(3.043931106\)
\(L(\frac12)\) \(\approx\) \(3.043931106\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−19.93207313242659, −19.12556272518370, −18.06720294101619, −17.43744099797714, −16.57244738891537, −15.80417244911308, −14.95807309942630, −14.39578789110842, −13.70228878820388, −13.09402965741118, −12.16006301979351, −11.46211301801455, −10.50540226217556, −9.450916859891261, −8.958168635991185, −7.695368349758993, −6.838927692088661, −6.040430141035949, −4.832478418603333, −3.967184441932551, −2.813745287668894, −1.671507921847687, 1.671507921847687, 2.813745287668894, 3.967184441932551, 4.832478418603333, 6.040430141035949, 6.838927692088661, 7.695368349758993, 8.958168635991185, 9.450916859891261, 10.50540226217556, 11.46211301801455, 12.16006301979351, 13.09402965741118, 13.70228878820388, 14.39578789110842, 14.95807309942630, 15.80417244911308, 16.57244738891537, 17.43744099797714, 18.06720294101619, 19.12556272518370, 19.93207313242659

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